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Evokation
  
 
Index
  

 

 

-
-
-
-
-
-
-
-
-
-
1
2
3
4
5
6
7
8
9
A
=
1
-
5
ADDED
18
18
9
-
-
-
-
-
-
-
-
-
9
T
=
2
-
2
TO
35
8
8
-
-
-
-
-
-
-
-
8
-
A
=
1
-
3
ALL
25
7
7
-
-
-
-
-
-
-
7
-
-
M
=
4
-
5
MINUS
76
22
4
-
-
-
-
4
-
-
-
-
-
N
=
5
-
4
NONE
48
21
3
-
-
-
3
-
-
-
-
-
-
S
=
1
-
6
SHARED
55
28
1
-
1
-
-
-
-
-
-
-
-
B
=
2
-
2
BY
27
9
9
-
-
-
-
-
-
-
-
-
9
E
=
5
-
10
EVERYTHING
133
61
7
-
-
-
-
-
-
-
7
-
-
M
=
4
-
10
MULTIPLED
121
49
4
-
-
-
-
4
-
-
-
-
-
I
=
9
-
2
IN
23
14
5
-
-
-
-
-
5
-
-
-
-
A
=
1
-
9
ABUNDANCE
65
29
2
-
-
2
-
-
-
-
-
-
-
-
-
35
-
57
First Total
626
266
59
-
1
2
3
8
5
6
14
8
18
-
-
3+5
-
5+7
Add to Reduce
6+2+6
2+6+6
5+9
-
-
-
-
-
-
-
1+4
-
1+8
-
-
8
-
12
Second Total
14
14
14
-
1
2
3
8
5
6
5
8
9
-
-
-
-
1+2
Reduce to Deduce
1+4
1+4
1+4
-
-
-
-
-
-
-
-
-
-
-
-
8
-
3
Essence of Number
5
5
5
-
1
2
3
8
5
6
5
8
9

 

 

26
A
B
C
D
E
F
G
H
I
J
K
L
M
N
O
P
Q
R
S
T
U
V
W
X
Y
Z
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
8
9
-
-
-
-
5
6
-
-
-
1
-
-
-
-
6
-
8
+
=
43
4+3
=
7
=
7
=
7
-
-
-
-
-
-
-
-
8
9
-
-
-
-
14
15
-
-
-
19
-
-
-
-
24
-
26
+
=
115
1+1+5
=
7
=
7
=
7
26
A
B
C
D
E
F
G
H
I
J
K
L
M
N
O
P
Q
R
S
T
U
V
W
X
Y
Z
-
-
-
-
-
-
-
-
-
-
-
1
2
3
4
5
6
7
-
-
1
2
3
4
-
-
7
8
9
-
2
3
4
5
-
7
-
+
=
83
8+3
=
11
1+1
2
=
2
-
1
2
3
4
5
6
7
-
-
10
11
12
13
-
-
16
17
18
-
20
21
22
23
-
25
-
+
=
236
2+3+6
=
11
1+1
2
=
2
26
A
B
C
D
E
F
G
H
I
J
K
L
M
N
O
P
Q
R
S
T
U
V
W
X
Y
Z
-
-
-
-
-
-
-
-
-
-
-
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
+
=
351
3+5+1
=
9
=
9
=
9
-
1
2
3
4
5
6
7
8
9
1
2
3
4
5
6
7
8
9
1
2
3
4
5
6
7
8
+
=
126
1+2+6
=
9
=
9
=
9
26
A
B
C
D
E
F
G
H
I
J
K
L
M
N
O
P
Q
R
S
T
U
V
W
X
Y
Z
-
-
-
-
-
-
-
-
-
-
-
1
-
-
-
-
-
-
-
-
1
-
-
-
-
-
-
-
-
1
-
-
-
-
-
-
-
+
=
1
occurs
x
3
=
3
=
3
-
-
2
-
-
-
-
-
-
-
-
2
-
-
-
-
-
-
-
-
2
-
-
-
-
-
-
+
=
2
occurs
x
3
=
6
=
6
-
-
-
3
-
-
-
-
-
-
-
-
3
-
-
-
-
-
-
-
-
3
-
-
-
-
-
+
=
3
occurs
x
3
=
9
=
9
-
-
-
-
4
-
-
-
-
-
-
-
-
4
-
-
-
-
-
-
-
-
4
-
-
-
-
+
=
4
occurs
x
3
=
12
1+2
3
-
-
-
-
-
5
-
-
-
-
-
-
-
-
5
-
-
-
-
-
-
-
-
5
-
-
-
+
=
5
occurs
x
3
=
15
1+5
6
-
-
-
-
-
-
6
-
-
-
-
-
-
-
-
6
-
-
-
-
-
-
-
-
6
-
-
+
=
6
occurs
x
3
=
18
1+8
9
-
-
-
-
-
-
-
7
-
-
-
-
-
-
-
-
7
-
-
-
-
-
-
-
-
7
-
+
=
7
occurs
x
3
=
21
2+1
3
-
-
-
-
-
-
-
-
8
-
-
-
-
-
-
-
-
8
-
-
-
-
-
-
-
-
8
+
=
8
occurs
x
3
=
24
2+4
6
-
-
-
-
-
-
-
-
-
9
-
-
-
-
-
-
-
-
9
-
-
-
-
-
-
-
-
+
=
9
occurs
x
2
=
18
1+8
9
26
A
B
C
D
E
F
G
H
I
J
K
L
M
N
O
P
Q
R
S
T
U
V
W
X
Y
Z
-
-
45
-
-
26
-
126
-
54
-
-
-
-
-
-
-
-
-
9
-
-
-
-
-
-
-
-
9
-
-
-
-
-
-
-
-
-
-
4+5
-
-
2+6
-
1+2+6
-
5+4
26
A
B
C
D
E
F
G
H
I
J
K
L
M
N
O
P
Q
R
S
T
U
V
W
X
Y
Z
-
-
9
-
-
8
-
9
-
9
-
1
2
3
4
5
6
7
8
9
1
2
3
4
5
6
7
8
9
1
2
3
4
5
6
7
8
-
-
-
-
-
-
-
-
-
-
26
A
B
C
D
E
F
G
H
I
J
K
L
M
N
O
P
Q
R
S
T
U
V
W
X
Y
Z
-
-
9
-
-
8
-
9
-
9

 

 

-
SIMONIDES
-
-
-
1
S
19
1
1
4
O+M+N+I
51
24
6
4
D+I+E+S
37
19
1
9
SIMONIDES
107
44
35
-
-
1+0+7
4+4
3+5
9
SIMONIDES
8
8
8

 

 

-
SIMONIDES
-
-
-
1
S
19
10
1
1
I
9
9
9
3
M+O+N
42
15
6
1
I
9
9
9
2
D+E
9
9
9
1
S
19
10
1
9
SIMONIDES
107
62
35
-
-
1+0+7
6+2
3+5
9
SIMONIDES
8
8
8

 

 

-
9
S
I
M
O
N
I
D
E
S
-
-
-
-
-
-
-
-
-
-
-
-
1
9
-
6
5
9
-
-
1
+
=
31
3+1
=
4
=
4
=
4
-
-
19
9
-
15
14
9
-
-
19
+
=
85
8+5
=
13
1+3
4
=
4
-
9
S
I
M
O
N
I
D
E
S
-
-
-
-
-
-
-
-
-
-
-
-
-
-
4
-
-
-
4
5
-
+
=
13
1+3
=
4
=
4
=
4
-
-
-
-
13
-
-
-
4
5
-
+
=
22
2+2
=
4
=
4
=
4
-
9
S
I
M
O
N
I
D
E
S
-
-
-
-
-
-
-
-
-
-
-
-
19
9
13
15
14
9
4
5
19
+
=
107
1+0+7
=
6
=
6
=
6
-
-
1
9
4
6
5
9
4
5
1
+
=
44
2+4
=
6
=
6
=
6
-
9
S
I
M
O
N
I
D
E
S
-
-
-
-
-
-
-
-
-
-
-
-
1
-
-
-
-
-
-
-
1
-
-
1
occurs
x
2
=
2
=
2
2
-
-
-
-
-
-
-
-
-
-
-
-
2
TWO
2
-
-
-
-
-
3
-
-
-
-
-
-
-
-
-
-
-
-
3
THREE
3
-
-
-
-
-
-
-
-
-
4
-
-
-
4
-
-
-
-
4
occurs
x
2
=
8
=
8
-
-
-
-
-
-
5
-
-
5
-
-
-
5
occurs
x
2
=
10
1+0
1
-
-
-
-
-
6
-
-
-
-
-
-
-
6
occurs
x
1
=
6
=
6
7
-
-
-
-
-
-
-
-
-
-
-
-
7
SEVEN
7
-
-
-
-
-
8
-
-
-
-
-
-
-
-
-
-
-
-
8
EIGHT
8
-
-
-
-
-
-
-
-
9
-
-
-
9
-
-
-
-
-
9
occurs
x
2
=
18
1+8
9
20
9
S
I
M
O
N
I
D
E
S
-
-
25
-
-
9
-
44
-
26
2+0
-
-
-
-
-
-
-
-
-
-
-
-
2+5
-
-
-
-
4+4
-
2+6
2
9
S
I
M
O
N
I
D
E
S
-
-
7
-
-
9
-
8
-
8
-
-
1
9
4
6
5
9
4
5
1
-
-
-
-
-
-
-
-
-
-
2
9
S
I
M
O
N
I
D
E
S
-
-
7
-
-
9
-
8
-
8

 

 

9
S
I
M
O
N
I
D
E
S
-
-
-
-
-
-
-
-
-
-
-
1
9
-
6
5
9
-
-
1
+
=
31
3+1
=
4
=
4
=
4
-
19
9
-
15
14
9
-
-
19
+
=
85
8+5
=
13
1+3
4
=
4
9
S
I
M
O
N
I
D
E
S
-
-
-
-
-
-
-
-
-
-
-
-
-
4
-
-
-
4
5
-
+
=
13
1+3
=
4
=
4
=
4
-
-
-
13
-
-
-
4
5
-
+
=
22
2+2
=
4
=
4
=
4
9
S
I
M
O
N
I
D
E
S
-
-
-
-
-
-
-
-
-
-
-
19
9
13
15
14
9
4
5
19
+
=
107
1+0+7
=
6
=
6
=
6
-
1
9
4
6
5
9
4
5
1
+
=
44
2+4
=
6
=
6
=
6
9
S
I
M
O
N
I
D
E
S
-
-
-
-
-
-
-
-
-
-
-
1
-
-
-
-
-
-
-
1
-
-
1
occurs
x
2
=
2
=
2
-
-
-
4
-
-
-
4
-
-
-
-
4
occurs
x
2
=
8
=
8
-
-
-
-
-
5
-
-
5
-
-
-
5
occurs
x
2
=
10
1+0
1
-
-
-
-
6
-
-
-
-
-
-
-
6
occurs
x
1
=
6
=
6
-
-
9
-
-
-
9
-
-
-
-
-
9
occurs
x
2
=
18
1+8
9
9
S
I
M
O
N
I
D
E
S
-
-
25
-
-
9
-
44
-
26
-
-
-
-
-
-
-
-
-
-
-
-
2+5
-
-
-
-
4+4
-
2+6
9
S
I
M
O
N
I
D
E
S
-
-
7
-
-
9
-
8
-
8
-
1
9
4
6
5
9
4
5
1
-
-
-
-
-
-
-
-
-
-
9
S
I
M
O
N
I
D
E
S
-
-
7
-
-
9
-
8
-
8

 

SIMONIDES

 

S MON DES

 

7
S
M
O
N
D
E
S
-
-
-
-
-
-
-
-
-
-
-
1
-
6
5
-
-
1
+
=
13
1+3
=
4
=
4
=
4
-
19
-
15
14
-
-
19
+
=
67
6+7
=
13
1+3
4
=
4
7
S
M
O
N
D
E
S
-
-
-
-
-
-
-
-
-
-
-
-
4
-
-
4
5
-
+
=
13
1+3
=
4
=
4
=
4
-
-
13
-
-
4
5
-
+
=
22
2+2
=
4
=
4
=
4
7
S
M
O
N
D
E
S
-
-
-
-
-
-
-
-
-
-
-
19
13
15
14
4
5
19
+
=
89
8+9
=
17
1+7
8
=
8
-
1
4
6
5
4
5
1
+
=
26
2+6
=
8
=
8
=
8
7
S
M
O
N
D
E
S
-
-
-
-
-
-
-
-
-
-
-
1
-
-
-
-
-
1
-
-
1
occurs
x
2
=
2
=
2
-
-
4
-
-
4
-
-
-
-
4
occurs
x
2
=
8
=
8
-
-
-
-
5
-
5
-
-
-
5
occurs
x
2
=
10
1+0
1
-
-
-
6
-
-
-
-
-
-
6
occurs
x
1
=
6
=
6
7
S
M
O
N
D
E
S
-
-
16
-
-
7
-
26
-
17
-
-
-
-
-
-
-
-
-
-
1+6
-
-
-
-
2+6
-
1+7
7
S
M
O
N
D
E
S
-
-
7
-
-
7
-
8
-
8
-
1
4
6
5
4
5
1
-
-
-
-
-
-
-
-
-
-
7
S
M
O
N
D
E
S
-
-
7
-
-
7
-
8
-
8

 

 

9
SIMONIDES
107
44
8

 

 

9
SIMONIDES
107
44
8
7
MELICUS
82
28
1
16
-
189
72
35
1+6
-
1+8+9
7+2
3+5
7
-
18
9
9
-
-
1+8
-
-
7
-
9
9
9

 

 

-
16
S
I
M
O
N
I
D
E
S
-
M
E
L
I
C
U
S
-
-
-
-
-
-
-
-
-
-
-
-
1
9
-
6
5
9
-
-
1
-
-
-
-
9
-
-
1
+
=
41
4+1
=
5
=
5
=
5
-
-
19
9
-
15
14
9
-
-
19
-
-
-
-
9
-
-
19
+
=
113
1+1+3
=
5
=
5
=
5
-
9
S
I
M
O
N
I
D
E
S
-
M
E
L
I
C
U
S
-
-
-
-
-
-
-
-
-
-
-
-
-
-
4
-
-
-
4
5
-
-
4
5
3
-
3
3
-
+
=
31
1+3
=
4
=
4
=
4
-
-
-
-
13
-
-
-
4
5
-
-
13
5
12
-
3
21
-
+
=
76
7+6
=
13
1+3
4
=
4
-
16
S
I
M
O
N
I
D
E
S
-
M
E
L
I
C
U
S
-
-
-
-
-
-
-
-
-
-
-
-
19
9
13
15
14
9
4
5
19
-
13
5
12
9
3
21
19
+
=
189
1+8+9
=
1+8
=
9
=
9
-
-
1
9
4
6
5
9
4
5
1
-
4
5
3
9
3
3
1
+
=
72
7+2
=
9
=
9
=
9
-
16
S
I
M
O
N
I
D
E
S
-
M
E
L
I
C
U
S
-
-
-
-
-
-
-
-
-
-
-
-
1
-
-
-
-
-
-
-
1
-
-
-
-
-
-
-
1
-
-
1
occurs
x
3
=
3
=
3
2
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
2
TWO
2
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
3
-
3
3
-
-
-
3
occurs
x
3
=
9
=
9
-
-
-
-
4
-
-
-
4
-
-
-
4
-
-
-
-
-
-
-
-
4
occurs
x
3
=
12
1+2
3
-
-
-
-
-
-
5
-
-
5
-
-
-
5
-
-
-
-
-
-
-
5
occurs
x
3
=
15
1+5
6
-
-
-
-
-
6
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
6
occurs
x
1
=
6
=
6
7
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
7
SEVEN
7
-
-
-
-
-
8
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
8
EIGHT
8
-
-
-
-
-
-
-
-
9
-
-
-
9
-
-
-
-
-
-
-
9
-
-
-
-
-
9
occurs
x
3
=
27
2+7
9
17
16
S
I
M
O
N
I
D
E
S
-
M
E
L
I
C
U
S
-
-
28
-
-
16
-
72
-
36
1+7
1+6
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
2+8
-
-
1+6
-
7+2
-
3+6
8
7
S
I
M
O
N
I
D
E
S
-
M
E
L
I
C
U
S
-
-
10
-
-
7
-
9
-
9
-
-
1
9
4
6
5
9
4
5
1
-
4
5
3
9
3
3
1
-
-
1+0
-
-
-
-
-
-
-
8
7
S
I
M
O
N
I
D
E
S
-
M
E
L
I
C
U
S
-
-
1
-
-
7
-
9
-
9

 

 

6
S
I
M
O
N
I
D
E
S
-
M
E
L
I
C
U
S
-
-
-
-
-
-
-
-
-
-
-
1
9
-
6
5
9
-
-
1
-
-
-
-
9
-
-
1
+
=
41
4+1
=
5
=
5
=
5
-
19
9
-
15
14
9
-
-
19
-
-
-
-
9
-
-
19
+
=
113
1+1+3
=
5
=
5
=
5
9
S
I
M
O
N
I
D
E
S
-
M
E
L
I
C
U
S
-
-
-
-
-
-
-
-
-
-
-
-
-
4
-
-
-
4
5
-
-
4
5
3
-
3
3
-
+
=
31
1+3
=
4
=
4
=
4
-
-
-
13
-
-
-
4
5
-
-
13
5
12
-
3
21
-
+
=
76
7+6
=
13
1+3
4
=
4
16
S
I
M
O
N
I
D
E
S
-
M
E
L
I
C
U
S
-
-
-
-
-
-
-
-
-
-
-
19
9
13
15
14
9
4
5
19
-
13
5
12
9
3
21
19
+
=
189
1+8+9
=
1+8
=
9
=
9
-
1
9
4
6
5
9
4
5
1
-
4
5
3
9
3
3
1
+
=
72
7+2
=
9
=
9
=
9
16
S
I
M
O
N
I
D
E
S
-
M
E
L
I
C
U
S
-
-
-
-
-
-
-
-
-
-
-
1
-
-
-
-
-
-
-
1
-
-
-
-
-
-
-
1
-
-
1
occurs
x
3
=
3
=
3
-
-
-
-
-
-
-
-
-
-
-
-
-
3
-
3
3
-
-
-
3
occurs
x
3
=
9
=
9
-
-
-
4
-
-
-
4
-
-
-
4
-
-
-
-
-
-
-
-
4
occurs
x
3
=
12
1+2
3
-
-
-
-
-
5
-
-
5
-
-
-
5
-
-
-
-
-
-
-
5
occurs
x
3
=
15
1+5
6
-
-
-
-
6
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
6
occurs
x
1
=
6
=
6
-
-
9
-
-
-
9
-
-
-
-
-
-
-
9
-
-
-
-
-
9
occurs
x
3
=
27
2+7
9
16
S
I
M
O
N
I
D
E
S
-
M
E
L
I
C
U
S
-
-
28
-
-
16
-
72
-
36
1+6
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
2+8
-
-
1+6
-
7+2
-
3+6
7
S
I
M
O
N
I
D
E
S
-
M
E
L
I
C
U
S
-
-
10
-
-
7
-
9
-
9
-
1
9
4
6
5
9
4
5
1
-
4
5
3
9
3
3
1
-
-
1+0
-
-
-
-
-
-
-
7
S
I
M
O
N
I
D
E
S
-
M
E
L
I
C
U
S
-
-
1
-
-
7
-
9
-
9

 

 

6
S
I
M
O
N
I
D
E
S
M
E
L
I
C
U
S
-
-
-
-
-
-
-
-
-
-
-
1
9
-
6
5
9
-
-
1
-
-
-
9
-
-
1
+
=
41
4+1
=
5
=
5
=
5
-
19
9
-
15
14
9
-
-
19
-
-
-
9
-
-
19
+
=
113
1+1+3
=
5
=
5
=
5
9
S
I
M
O
N
I
D
E
S
M
E
L
I
C
U
S
-
-
-
-
-
-
-
-
-
-
-
-
-
4
-
-
-
4
5
-
4
5
3
-
3
3
-
+
=
31
1+3
=
4
=
4
=
4
-
-
-
13
-
-
-
4
5
-
13
5
12
-
3
21
-
+
=
76
7+6
=
13
1+3
4
=
4
16
S
I
M
O
N
I
D
E
S
M
E
L
I
C
U
S
-
-
-
-
-
-
-
-
-
-
-
19
9
13
15
14
9
4
5
19
13
5
12
9
3
21
19
+
=
189
1+8+9
=
1+8
=
9
=
9
-
1
9
4
6
5
9
4
5
1
4
5
3
9
3
3
1
+
=
72
7+2
=
9
=
9
=
9
16
S
I
M
O
N
I
D
E
S
M
E
L
I
C
U
S
-
-
-
-
-
-
-
-
-
-
-
1
-
-
-
-
-
-
-
1
-
-
-
-
-
-
1
-
-
1
occurs
x
3
=
3
=
3
-
-
-
-
-
-
-
-
-
-
-
-
3
-
3
3
-
-
-
3
occurs
x
3
=
9
=
9
-
-
-
4
-
-
-
4
-
-
4
-
-
-
-
-
-
-
-
4
occurs
x
3
=
12
1+2
3
-
-
-
-
-
5
-
-
5
-
-
5
-
-
-
-
-
-
-
5
occurs
x
3
=
15
1+5
6
-
-
-
-
6
-
-
-
-
-
-
-
-
-
-
-
-
-
-
6
occurs
x
1
=
6
=
6
-
-
9
-
-
-
9
-
-
-
-
-
-
9
-
-
-
-
-
9
occurs
x
3
=
27
2+7
9
16
S
I
M
O
N
I
D
E
S
M
E
L
I
C
U
S
-
-
28
-
-
16
-
72
-
36
1+6
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
2+8
-
-
1+6
-
7+2
-
3+6
7
S
I
M
O
N
I
D
E
S
M
E
L
I
C
U
S
-
-
10
-
-
7
-
9
-
9
-
1
9
4
6
5
9
4
5
1
4
5
3
9
3
3
1
-
-
1+0
-
-
-
-
-
-
-
7
S
I
M
O
N
I
D
E
S
M
E
L
I
C
U
S
-
-
1
-
-
7
-
9
-
9

 

 

20
DE OCCULTA PHILOSOPHIA
212
95
5
-
-
-
-
-
-
-
-
-
-
2
DE
9
9
9
7
OCCULTA
75
21
3
11
PHILOSOPHIA
128
65
2
20
DE OCCULTA PHILOSOPHIA
212
95
14
2+0
-
2+1+2
9+5
1+4
2
DE OCCULTA PHILOSOPHIA
5
14
5
-
-
-
1+4
-
2
DE OCCULTA PHILOSOPHIA
5
5
5

 

 

-
20
D
E
-
O
C
C
U
L
T
A
-
P
H
I
L
O
S
O
P
H
I
A
-
-
-
-
-
-
-
-
-
-
53
-
-
-
-
6
-
-
-
-
-
-
-
-
8
9
-
6
1
6
-
8
9
-
+
=
53
5+3
=
8
=
8
=
8
98
-
-
-
-
15
-
-
-
-
-
-
-
-
8
9
-
15
19
15
-
8
9
-
+
=
98
9+8
=
17
1+7
8
=
8
-
20
D
E
-
O
C
C
U
L
T
A
-
P
H
I
L
O
S
O
P
H
I
A
-
-
-
-
-
-
-
-
-
-
42
-
4
5
-
-
3
3
3
3
2
1
-
7
-
-
3
-
-
-
7
-
-
1
+
=
42
4+2
=
6
=
6
=
6
114
-
4
5
-
-
3
3
21
12
20
1
-
16
-
-
12
-
-
-
16
-
-
1
+
=
114
1+1+4
=
6
=
6
=
6
-
20
D
E
-
O
C
C
U
L
T
A
`
P
H
I
L
O
S
O
P
H
I
A
-
-
-
-
-
-
-
-
-
-
212
-
4
5
-
15
3
3
21
12
20
1
-
16
8
9
12
15
19
15
16
8
9
1
+
=
212
2+1+2
=
5
=
5
=
5
95
-
4
5
-
6
3
3
3
3
2
1
-
7
8
9
3
6
1
6
7
8
9
1
+
=
95
9+5
=
14
1+4
5
=
5
-
20
D
E
-
O
C
C
U
L
T
A
-
P
H
I
L
O
S
O
P
H
I
A
-
-
-
-
-
-
-
-
-
-
1
-
-
-
-
-
-
-
-
-
-
1
-
-
-
-
-
-
1
-
-
-
-
1
-
-
1
occurs
x
3
=
3
=
3
2
-
-
-
-
-
-
-
-
-
2
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
2
occurs
x
1
=
2
=
2
3
-
-
-
-
-
3
3
3
3
-
-
-
-
-
-
3
-
-
-
-
-
-
-
-
-
3
occurs
x
5
=
15
1+5
6
4
-
4
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
4
occurs
x
1
=
4
=
4
5
-
-
5
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
5
occurs
x
1
=
5
=
5
6
-
-
-
-
6
-
-
-
-
-
-
-
-
-
-
-
6
-
6
-
-
-
-
-
-
6
occurs
x
3
=
18
1+8
9
7
-
-
-
-
-
-
-
-
-
-
-
-
7
-
-
-
-
-
-
7
-
-
-
-
-
7
occurs
x
2
=
14
1+4
5
8
-
-
-
-
-
-
-
-
-
-
-
-
-
8
-
-
-
-
-
-
8
-
-
-
-
8
occurs
x
2
=
16
1+6
7
9
-
-
-
-
-
-
-
-
-
-
-
-
-
-
9
-
-
-
-
-
-
9
-
-
-
9
occurs
x
2
=
18
1+8
9
45
20
D
E
-
O
C
C
U
L
T
A
-
P
H
I
L
O
S
O
P
H
I
A
-
-
45
-
-
20
-
95
-
50
4+5
-
-
-
-
3
3
3
3
-
-
-
-
-
3
-
-
-
-
-
-
-
9
-
-
-
4+5
-
-
2+0
-
9+5
-
5+0
9
20
D
E
-
O
C
C
U
L
T
A
-
P
H
I
L
O
S
O
P
H
I
A
-
-
9
-
-
2
-
14
-
5
-
-
4
5
-
6
3
3
3
3
2
1
-
7
8
9
3
6
1
6
7
8
9
1
-
-
-
-
-
-
-
1+4
-
-
9
20
D
E
-
O
C
C
U
L
T
A
-
P
H
I
L
O
S
O
P
H
I
A
-
-
9
-
-
2
-
5
-
5

 

 

-
20
D
E
O
C
C
U
L
T
A
P
H
I
L
O
S
O
P
H
I
A
-
-
-
-
-
-
-
-
-
-
53
-
-
-
6
-
-
-
-
-
-
-
8
9
-
6
1
6
-
8
9
-
+
=
53
5+3
=
8
=
8
=
8
98
-
-
-
15
-
-
-
-
-
-
-
8
9
-
15
19
15
-
8
9
-
+
=
98
9+8
=
17
1+7
8
=
8
-
20
D
E
O
C
C
U
L
T
A
P
H
I
L
O
S
O
P
H
I
A
-
-
-
-
-
-
-
-
-
-
42
-
4
5
-
3
3
3
3
2
1
7
-
-
3
-
-
-
7
-
-
1
+
=
42
4+2
=
6
=
6
=
6
114
-
4
5
-
3
3
21
12
20
1
16
-
-
12
-
-
-
16
-
-
1
+
=
114
1+1+4
=
6
=
6
=
6
-
20
D
E
O
C
C
U
L
T
A
P
H
I
L
O
S
O
P
H
I
A
-
-
-
-
-
-
-
-
-
-
212
-
4
5
15
3
3
21
12
20
1
16
8
9
12
15
19
15
16
8
9
1
+
=
212
2+1+2
=
5
=
5
=
5
95
-
4
5
6
3
3
3
3
2
1
7
8
9
3
6
1
6
7
8
9
1
+
=
95
9+5
=
14
1+4
5
=
5
-
20
D
E
O
C
C
U
L
T
A
P
H
I
L
O
S
O
P
H
I
A
-
-
-
-
-
-
-
-
-
-
1
-
-
-
-
-
-
-
-
-
1
-
-
-
-
-
1
-
-
-
-
1
-
-
1
occurs
x
3
=
3
=
3
2
-
-
-
-
-
-
-
-
2
-
-
-
-
-
-
-
-
-
-
-
-
-
-
2
occurs
x
1
=
2
=
2
3
-
-
-
-
3
3
3
3
-
-
-
-
-
3
-
-
-
-
-
-
-
-
-
3
occurs
x
5
=
15
1+5
6
4
-
4
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
4
occurs
x
1
=
4
=
4
5
-
-
5
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
5
occurs
x
1
=
5
=
5
6
-
-
-
6
-
-
-
-
-
-
-
-
-
-
6
-
6
-
-
-
-
-
-
6
occurs
x
3
=
18
1+8
9
7
-
-
-
-
-
-
-
-
-
-
7
-
-
-
-
-
-
7
-
-
-
-
-
7
occurs
x
2
=
14
1+4
5
8
-
-
-
-
-
-
-
-
-
-
-
8
-
-
-
-
-
-
8
-
-
-
-
8
occurs
x
2
=
16
1+6
7
9
-
-
-
-
-
-
-
-
-
-
-
-
9
-
-
-
-
-
-
9
-
-
-
9
occurs
x
2
=
18
1+8
9
45
20
D
E
O
C
C
U
L
T
A
P
H
I
L
O
S
O
P
H
I
A
-
-
45
-
-
20
-
95
-
50
4+5
-
-
-
-
3
3
3
3
-
-
-
-
-
3
-
-
-
-
-
-
-
-
-
4+5
-
-
2+0
-
9+5
-
5+0
9
20
D
E
O
C
C
U
L
T
A
P
H
I
L
O
S
O
P
H
I
A
-
-
9
-
-
2
-
14
-
5
-
-
4
5
6
3
3
3
3
2
1
7
8
9
3
6
1
6
7
8
9
1
-
-
-
-
-
-
-
1+4
-
-
9
20
D
E
O
C
C
U
L
T
A
P
H
I
L
O
S
O
P
H
I
A
-
-
9
-
-
2
-
5
-
5

 

 

12
THE E AT DELPHI
113
59
5
-
-
-
-
-
-
-
-
-
-
8
PLUTARCH
99
36
9
-
-
-
-
-
-
-
-
-
-
2
DE
9
9
9
5
ISIDE
46
28
1
2
ET
25
7
7
7
OSIRIDE
79
43
7
16
DE ISIDE ET OSIRIDE
159
87
24
1+6
-
1+5+9
8+7
2+4
7
DE ISIDE ET OSIRIDE
15
15
6
-
-
1+5
1+5
-
7
DE ISIDE ET OSIRIDE
6
6
6

 

 

-
16
D
E
-
I
S
I
D
E
-
E
T
-
O
S
I
R
I
D
E
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
9
1
9
-
-
-
-
-
-
6
1
9
-
9
-
-
+
=
44
4+4
=
8
=
8
=
8
-
-
-
-
-
9
19
9
-
-
-
-
-
-
15
19
9
-
9
-
-
+
=
89
8+9
=
17
1+7
8
=
8
-
16
D
E
-
I
S
I
D
E
-
E
T
-
O
S
I
R
I
D
E
-
-
-
-
-
-
-
-
-
-
-
-
4
5
-
-
-
-
4
5
-
5
2
-
-
-
-
9
-
4
5
+
=
43
4+3
=
7
=
7
=
7
-
-
4
5
-
-
-
-
4
5
-
5
20
-
-
-
-
18
-
4
5
+
=
70
7+0
=
7
=
7
=
7
-
16
D
E
-
I
S
I
D
E
`
E
T
-
O
S
I
R
I
D
E
-
-
-
-
-
-
-
-
-
-
-
-
4
5
-
9
19
9
4
5
-
5
20
-
15
19
9
18
9
4
5
+
=
159
1+5+9
=
15
1+5
6
=
6
-
-
4
5
-
9
1
9
4
5
-
5
2
-
6
1
9
9
9
4
5
+
=
87
8+7
=
15
1+5
6
=
6
-
16
D
E
-
I
S
I
D
E
-
E
T
-
O
S
I
R
I
D
E
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
1
-
-
-
-
-
-
-
-
1
-
-
-
-
-
-
-
1
occurs
x
2
=
2
=
2
-
-
-
-
-
-
-
-
-
-
-
-
2
-
-
-
-
-
-
-
-
-
-
2
occurs
x
1
=
2
=
2
3
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
3
THREE
3
-
-
-
-
-
-
-
4
-
-
-
-
-
4
-
-
-
-
-
-
-
-
-
-
4
-
-
-
4
occurs
x
3
=
12
1+2
3
-
-
-
5
-
-
-
-
-
5
-
5
-
-
-
-
-
-
-
-
5
-
-
5
occurs
x
4
=
20
2+0
2
-
-
-
-
-
-
-
-
-
-
-
-
-
-
6
-
-
-
-
-
-
-
-
6
occurs
x
1
=
6
=
6
7
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
7
SEVEN
7
-
-
-
-
-
8
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
8
EIGHT
8
-
-
-
-
-
-
-
-
-
-
9
-
9
-
-
-
-
-
-
-
-
9
9
9
-
-
-
-
9
occurs
x
5
=
45
4+5
9
18
16
D
E
-
I
S
I
D
E
-
E
T
-
O
S
I
R
I
D
E
-
-
27
-
-
16
-
87
-
24
1+8
-
-
-
-
9
-
9
-
-
-
-
-
-
-
-
9
9
9
-
-
-
-
2+7
-
-
1+6
-
8+7
-
2+4
9
16
D
E
-
I
S
I
D
E
-
E
T
-
O
S
I
R
I
D
E
-
-
9
-
-
7
-
15
-
6
-
-
4
5
-
9
1
9
4
5
-
5
2
-
6
1
9
9
9
4
5
-
-
-
-
-
-
-
1+5
-
-
9
16
D
E
-
I
S
I
D
E
-
E
T
-
O
S
I
R
I
D
E
-
-
9
-
-
7
-
6
-
6

 

 

16
D
E
-
I
S
I
D
E
-
E
T
-
O
S
I
R
I
D
E
-
-
-
-
-
-
-
-
-
-
-
-
-
-
9
1
9
-
-
-
-
-
-
6
1
9
-
9
-
-
+
=
44
4+4
=
8
=
8
=
8
-
-
-
-
9
19
9
-
-
-
-
-
-
15
19
9
-
9
-
-
+
=
89
8+9
=
17
1+7
8
=
8
16
D
E
-
I
S
I
D
E
-
E
T
-
O
S
I
R
I
D
E
-
-
-
-
-
-
-
-
-
-
-
4
5
-
-
-
-
4
5
-
5
2
-
-
-
-
9
-
4
5
+
=
43
4+3
=
7
=
7
=
7
-
4
5
-
-
-
-
4
5
-
5
20
-
-
-
-
18
-
4
5
+
=
70
7+0
=
7
=
7
=
7
16
D
E
-
I
S
I
D
E
`
E
T
-
O
S
I
R
I
D
E
-
-
-
-
-
-
-
-
-
-
-
4
5
-
9
19
9
4
5
-
5
20
-
15
19
9
18
9
4
5
+
=
159
1+5+9
=
15
1+5
6
=
6
-
4
5
-
9
1
9
4
5
-
5
2
-
6
1
9
9
9
4
5
+
=
87
8+7
=
15
1+5
6
=
6
16
D
E
-
I
S
I
D
E
-
E
T
-
O
S
I
R
I
D
E
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
1
-
-
-
-
-
-
-
-
1
-
-
-
-
-
-
-
1
occurs
x
2
=
2
=
2
-
-
-
-
-
-
-
-
-
-
-
2
-
-
-
-
-
-
-
-
-
-
2
occurs
x
1
=
2
=
2
-
4
-
-
-
-
-
4
-
-
-
-
-
-
-
-
-
-
4
-
-
-
4
occurs
x
3
=
12
1+2
3
-
-
5
-
-
-
-
-
5
-
5
-
-
-
-
-
-
-
-
5
-
-
5
occurs
x
4
=
20
2+0
2
-
-
-
-
-
-
-
-
-
-
-
-
-
6
-
-
-
-
-
-
-
-
6
occurs
x
1
=
6
=
6
-
-
-
-
9
-
9
-
-
-
-
-
-
-
-
9
9
9
-
-
-
-
9
occurs
x
5
=
45
4+5
9
16
D
E
-
I
S
I
D
E
-
E
T
-
O
S
I
R
I
D
E
-
-
27
-
-
16
-
87
-
24
-
-
-
-
9
-
9
-
-
-
-
-
-
-
-
9
9
9
-
-
-
-
2+7
-
-
1+6
-
8+7
-
2+4
16
D
E
-
I
S
I
D
E
-
E
T
-
O
S
I
R
I
D
E
-
-
9
-
-
7
-
15
-
6
-
4
5
-
9
1
9
4
5
-
5
2
-
6
1
9
9
9
4
5
-
-
-
-
-
-
-
1+5
-
-
16
D
E
-
I
S
I
D
E
-
E
T
-
O
S
I
R
I
D
E
-
-
9
-
-
7
-
6
-
6

 

 

16
D
E
I
S
I
D
E
E
T
S
I
R
I
D
E
-
-
-
-
-
--
-
-
-
-
-
-
-
9
1
9
-
-
-
-
6
1
9
-
9
-
-
+
=
44
4+4
=
8
=
8
=
8
-
-
-
9
19
9
-
-
-
-
15
19
9
-
9
-
-
+
=
89
8+9
=
17
1+7
8
=
8
16
D
E
I
S
I
D
E
E
T
O
S
I
R
I
D
E
-
-
-
-
-
--
-
-
-
-
-
4
5
-
-
-
4
5
5
2
-
-
-
9
-
4
5
+
=
43
4+3
=
7
=
7
=
7
-
4
5
-
-
-
4
5
5
20
-
-
-
18
-
4
5
+
=
70
7+0
=
7
=
7
=
7
16
D
E
I
S
I
D
E
E
T
O
S
I
R
I
D
E
-
-
-
-
-
--
-
-
-
-
-
4
5
9
19
9
4
5
5
20
15
19
9
18
9
4
5
+
=
159
1+5+9
=
15
1+5
6
=
6
-
4
5
9
1
9
4
5
5
2
6
1
9
9
9
4
5
+
=
87
8+7
=
15
1+5
6
=
6
16
D
E
I
S
I
D
E
E
T
O
S
I
R
I
D
E
-
-
-
-
-
-
-
-
-
-
-
-
-
-
1
-
-
-
-
-
-
1
-
-
-
-
-
-
-
1
occurs
x
2
=
2
=
2
-
-
-
-
-
-
-
-
-
2
-
-
-
-
-
-
-
-
-
2
occurs
x
1
=
2
=
2
-
4
-
-
-
-
4
-
-
-
-
-
-
-
-
4
-
-
-
4
occurs
x
3
=
12
1+2
3
-
-
5
-
-
-
-
5
5
-
-
-
-
-
-
-
5
-
-
5
occurs
x
4
=
20
2+0
2
-
-
-
-
-
-
-
-
-
-
6
-
-
-
-
-
-
-
-
6
occurs
x
1
=
6
=
6
-
-
-
9
-
9
-
-
-
-
-
-
9
9
9
-
-
-
-
9
occurs
x
5
=
45
4+5
9
16
D
E
I
S
I
D
E
E
T
O
S
I
R
I
D
E
-
-
27
-
-
16
-
87
-
24
-
-
-
9
-
9
-
-
-
-
-
-
9
9
9
-
-
-
-
2+7
-
-
1+6
-
8+7
-
2+4
16
D
E
I
S
I
D
E
E
T
O
S
I
R
I
D
E
-
-
9
-
-
7
-
15
-
6
-
4
5
9
1
9
4
5
5
2
6
1
9
9
9
4
5
-
-
-
-
-
-
-
1+5
-
-
16
D
E
I
S
I
D
E
E
T
O
S
I
R
I
D
E
-
-
9
-
-
7
-
6
-
6

 

 

6
OSIRIS
89
35
8
4
ISIS
56
20
2
3
SET
44
8
8
13
Add to Reduce
189
63
18
1+3
Reduce to Deduce
1+8
6+3
1+8
4
Essence of Number
9
9
9
5
HORUS
81
27
9
4
GODS
45
18
9
8
CREATORS
99
36
9

 

 

 

I

THAT

AM

CRUCIFIED

A

CROSS A CROSS

 

 

Macbeth 1 [iii]

Shakespeare,

"Thrice to thine and thrice to mine,

and thrice again to make up nine"

 Number 9

The Search for the Sigma code

Cecil Balmond

Page 45

"Three times three, the trinity of trinities, gains select status then as the doubling and resourcing of special power.

From ancient times number nine was seen as a full complement; it was the cup of special promise that brimmed over. "

 

3 x 3 = 9 = 3 x 3

 

 

Daily Mail, Thursday, December 15, 2011

ANSWERS TO CORRESPONDENTS

Compiled by Charles Legge

Page 57

At a formal dinner celebrating the opening of the Watford St Albans branch railway line in 1958 several toasts were drunk 'with three times three'. What does this mean?

THE Watford to St Albans branch line, known locally as the Abbey Flyer, opened on May 5, 1858, so the dinner must have been part of the centenary celebrations.

The 'three times three' is the traditional three cheers ('Hip, Hip, Hooray!' shouted three times).

The origin of the cheer is obscure but it's recorded in 1816 when John Quincy Adams, an envoy to the English court, wrote in his diary that he had attended a dinner given by the Lord Mayor in the Mansion House and that 'every toast was drunk standing with what they call three time three hip! hip! hip! and nine huzzahs'.

It is more usual now to proclaim only two hips and 'huzzah' has become 'hooray'. In the 17th and 18th centuries, three huzzahs - two short ones followed by a long sustained one - were given by the British infantry before a charge.

It has been suggested 'hurrah' or 'hooray' comes from old Turkic and that the Turks used to shout 'Ur, ah!' meaning 'Come on, hit!".

A less likely explanation is that 'hip' dates from the crusades and is an acronym of Hierosolyma est perdita -'Jerusalem is lost'. Equally unlikely is the idea that 'hurrah' is a corruption of the Slavonic hu-raj - paradise. In this scheme, 'Hip Hip Hooray!' would mean 'Jerusalem is lost; paradise here we come!'

Barry Saunders, Watford.

 

 

THE ART OF MEMORY

FRANCIS A. YATES 1966

THREE LATIN SOURCES FOR THE CLASSICAL ART OF MEMORY

Page 21

"The Ad Herennium was a well known and much used text in the Middle Ages when it had an immense prestige because it was thought to be by Cicero. It was therefore believed that the precepts for the artificial memory which it expounded had been drawn up by Tullius himself"

Page 32

"Though the medieval tradition which assigned the authorship of Ad Herennium to 'Tullius' was wrong in fact, it was not wrong in its inference that the art of memory was practised and recommended by Tullius."

 

 

-
TULLIUS
-
-
-
1
T
20
2
2
1
U
21
3
3
1
L
12
3
3
1
L
12
3
3
1
I
9
9
9
1
U
21
3
3
1
S
19
10
1
7
TULLIUS
114
24
12
-
-
1+1+4
2+4
1+2
7
TULLIUS
6
6
6

 

 

-
4
T
U
L
L
I
U
S
-
-
-
-
-
-
-
-
-
-
-
-
-
--
-
-
9
-
1
+
=
10
1+0
=
1
=
1
=
1
-
-
-
--
-
-
9
-
19
+
=
28
2+8
=
10
1+0
1
=
1
-
4
T
U
L
L
I
U
S
-
-
-
-
-
-
-
-
-
-
-
-
2
3
3
3
-
3
-
+
=
14
1+4
=
5
=
5
=
5
-
-
20
21
12
12
-
21
-
+
=
86
8+6
=
14
1+4
5
=
5
-
4
T
U
L
L
I
U
S
-
-
-
-
-
-
-
-
-
-
-
-
20
21
12
12
9
21
19
+
=
114
1+1+4
=
6
=
6
=
6
-
-
2
3
3
3
9
3
1
+
=
24
2+4
=
6
=
6
=
6
-
4
T
U
L
L
I
U
S
-
-
-
-
-
-
-
-
-
-
-
-
-
--
-
-
-
--
1
-
-
1
occurs
x
1
=
1
=
1
-
-
2
--
-
-
-
--
-
-
-
2
occurs
x
1
=
2
=
2
-
-
-
3
3
3
-
3
-
-
-
3
occurs
x
4
=
12
1+2
3
4
-
-
--
-
-
-
--
-
-
-
4
FOUR
4
-
-
-
-
-
5
-
-
--
-
-
-
--
-
-
-
5
FIVE
5
-
-
-
-
-
6
-
-
--
-
-
-
--
-
-
-
6
SIX
6
-
-
-
-
-
7
-
-
--
-
-
-
--
-
-
-
7
SEVEN
7
-
-
-
-
-
8
-
-
--
-
-
-
--
-
-
-
8
EIGHT
8
-
-
-
-
-
-
-
-
--
-
-
9
--
-
-
-
9
occurs
x
1
=
9
=
9
30
4
T
U
L
L
I
U
S
-
-
15
-
-
7
-
24
-
15
3+0
-
-
3
3
3
-
3
-
-
-
1+5
-
-
-
-
2+4
-
1+5
3
4
T
U
L
L
I
U
S
-
-
6
-
-
7
-
6
-
6
-
-
2
3
3
3
9
3
1
-
-
-
-
-
-
-
-
-
-
3
4
T
U
L
L
I
U
S
-
-
6
-
-
4
-
6
-
6

 

 

4
T
U
L
L
I
U
S
-
-
-
-
-
-
-
-
-
-
-
-
--
-
-
9
-
1
+
=
10
1+0
=
1
=
1
=
1
-
-
--
-
-
9
-
19
+
=
28
2+8
=
10
1+0
1
=
1
4
T
U
L
L
I
U
S
-
-
-
-
-
-
-
-
-
-
-
2
3
3
3
-
3
-
+
=
14
1+4
=
5
=
5
=
5
-
20
21
12
12
-
21
-
+
=
86
8+6
=
14
1+4
5
=
5
4
T
U
L
L
I
U
S
-
-
-
-
-
-
-
-
-
-
-
20
21
12
12
9
21
19
+
=
114
1+1+4
=
6
=
6
=
6
-
2
3
3
3
9
3
1
+
=
24
2+4
=
6
=
6
=
6
4
T
U
L
L
I
U
S
-
-
-
-
-
-
-
-
-
-
-
-
--
-
-
-
--
1
-
-
1
occurs
x
1
=
1
=
1
-
2
--
-
-
-
--
-
-
-
2
occurs
x
1
=
2
=
2
-
-
3
3
3
-
3
-
-
-
3
occurs
x
4
=
12
1+2
3
-
-
--
-
-
9
--
-
-
-
9
occurs
x
1
=
9
=
9
4
T
U
L
L
I
U
S
-
-
15
-
-
7
-
24
-
15
-
-
3
3
3
-
3
-
-
-
1+5
-
-
-
-
2+4
-
1+5
4
T
U
L
L
I
U
S
-
-
6
-
-
7
-
6
-
6
-
2
3
3
3
9
3
1
-
-
-
-
-
-
-
-
-
-
4
T
U
L
L
I
U
S
-
-
6
-
-
4
-
6
-
6

 

 

Page 282

"By the THE LADDER OF MINERVA we rise from the first to the last..."

 

 

THE ART OF MEMORY

FRANCIS A. YATES 1966

THREE LATIN SOURCES FOR THE CLASSICAL ART OF MEMORY

Page 175

CHAPTER 8

LULLISM AS AN ART OF MEMORY

THOUGH we have now reached the Renaissance, with Camillo, we have to retrace our steps to the Middle Ages during this chapter. For there was another kind of art of memory which began in the Middle Ages, which continued into the Renaissance and beyond, and which it was the aim of many in the Renaissance to combine with the classical art in some new synthesis whereby memory should reach still further heights of insight and of power. This other art of memory was the Art of Ramon Lull.

Lullism and its history is a most difficult subject and one for the exploration of which the full materials have not yet been assembled; The enormous number of Lull's own writings, some of them still unpublished, the vast Lullist literature written by his followers, the extreme complexity of Lullism, make it impossible as yet to reach very definite conclusions about what is, undoubtedly.ra strand of major importance in the European tradition. And what I have to do now is to write one not very long chapter giving some idea of what the Art of Ramon Lull was like, of why it was an art of memory, of how it differs from the classical art of memory, and of how Lullism became absorbed at the Renaissance into Renaissance forms of the classical art.

Obviously- I am attempting the impossible, yet the impossible must be attempted because it is essential for the later part of this book that there should be some sketch at this stage of Lullism itself. The chapter is based on my own two articles on the art of Ramon Lull:' it is orientated towards a comparison of Lullism as an art of memory with. the classical art; and it is not concerned solely with 'genuine' Lullism but also with the Renaissance interpretation of Lullism, for it is this. which is important for the next stages of our history.

 

ART AT R AT ART

 

 

-
LULL
-
-
-
1
L
12
3
3
1
U
21
3
3
1
L
12
3
3
1
L
12
3
3
4
LULL
57
12
12
-
-
5+7
1+2
1+2
4
LULL
12
3
3
-
-
1+2
-
-
4
LULL
3
3
3

 

 

-
4
L
U
L
L
-
-
-
-
-
-
-
-
-
-
-
-
3
3
3
3
+
=
12
1+2
=
3
=
3
=
3
-
-
12
21
12
12
+
=
57
5+7
=
12
1+2
3
=
3
-
4
L
U
L
L
-
-
-
-
-
-
-
-
-
-
-
-
3
3
3
3
+
=
12
1+2
=
3
=
3
=
3
-
-
12
21
12
12
+
=
57
5+7
=
12
1+2
3
=
3
-
4
L
U
L
L
-
-
-
-
-
-
-
-
-
-
-
-
12
21
12
12
+
=
57
5+7
=
12
=
3
=
3
-
-
3
3
3
3
+
=
12
1+2
=
3
=
3
=
3
-
4
L
U
L
L
-
-
-
-
-
-
-
-
-
-
1
-
-
--
-
-
-
-
1
ONE
1
-
-
-
-
-
2
-
-
--
-
-
-
-
2
TWO
2
-
-
-
-
-
-
-
3
3
3
3
-
-
3
occurs
x
4
=
12
1+2
3
4
-
-
--
-
-
-
-
4
FOUR
4
-
-
-
-
-
5
-
-
--
-
-
-
-
5
FIVE
5
-
-
-
-
-
6
-
-
--
-
-
-
-
6
SIX
6
-
-
-
-
-
7
-
-
--
-
-
-
-
7
SEVEN
7
-
-
-
-
-
8
-
-
--
-
-
-
-
8
EIGHT
8
-
-
-
-
-
9
-
-
--
-
-
-
-
9
NINE
9
-
-
-
-
-
42
4
L
U
L
L
-
-
3
-
-
4
-
12
-
3
4+2
-
3
3
3
3
-
-
-
-
-
-
-
1+2
-
-
6
4
L
U
L
L
-
-
3
-
-
4
-
3
-
3
-
-
3
3
3
3
-
-
-
-
-
-
-
-
-
-
6
4
L
U
L
L
-
-
3
-
-
4
-
3
-
3

 

 

-
4
L
U
L
L
-
-
-
-
-
-
-
-
-
-
-
-
3
3
3
3
+
=
12
1+2
=
3
=
3
=
3
-
-
12
21
12
12
+
=
57
5+7
=
12
1+2
3
=
3
-
4
L
U
L
L
-
-
-
-
-
-
-
-
-
-
-
-
3
3
3
3
+
=
12
1+2
=
3
=
3
=
3
-
-
12
21
12
12
+
=
57
5+7
=
12
1+2
3
=
3
-
4
L
U
L
L
-
-
-
-
-
-
-
-
-
-
-
-
12
21
12
12
+
=
57
5+7
=
12
=
3
=
3
-
-
3
3
3
3
+
=
12
1+2
=
3
=
3
=
3
-
4
L
U
L
L
-
-
-
-
-
-
-
-
-
-
-
-
3
3
3
3
-
-
3
occurs
x
4
=
12
1+2
3
42
4
L
U
L
L
-
-
3
-
-
4
-
12
-
3
4+2
-
3
3
3
3
-
-
-
-
-
-
-
1+2
-
-
6
4
L
U
L
L
-
-
3
-
-
4
-
3
-
3
-
-
3
3
3
3
-
-
-
-
-
-
-
-
-
-
6
4
L
U
L
L
-
-
3
-
-
4
-
3
-
3

 

 

-
4
L
U
L
L
-
-
-
-
-
-
-
-
-
-
-
-
12
21
12
12
+
=
57
5+7
=
12
=
3
=
3
-
-
3
3
3
3
+
=
12
1+2
=
3
=
3
=
3
-
4
L
U
L
L
-
-
-
-
-
-
-
-
-
-
-
-
3
3
3
3
-
-
3
occurs
x
4
=
12
1+2
3
42
4
L
U
L
L
-
-
3
-
-
4
-
12
-
3
4+2
-
3
3
3
3
-
-
-
-
-
-
-
1+2
-
-
6
4
L
U
L
L
-
-
3
-
-
4
-
3
-
3
-
-
3
3
3
3
-
-
-
-
-
-
-
-
-
-
6
4
L
U
L
L
-
-
3
-
-
4
-
3
-
3

 

 

THE ART OF MEMORY

FRANCIS A. YATES 1966

THREE LATIN SOURCES FOR THE CLASSICAL ART OF MEMORY

Page 111

MEDIEVAL MEMORY AND THE FORMATION OF IMAGES

"Petrarch makes his allusions to the art by introducing examples of men of antiquity famed for good memories and associating these with the classical art. His paragraph on the memories of Lucullus and Hortensius begins thus: - 'Memory is of two kinds, one for things, one for words.' 57

LUCULLUS 33333331 LUCULLUS

 

 

-
LUCULLUS
-
-
-
1
L
12
3
3
1
U
21
3
3
1
C
3
3
3
1
U
21
3
3
1
L
12
3
3
1
L
12
3
3
1
U
21
3
3
1
S
19
10
1
8
LUCULLUS
121
31
22
-
-
1+2+1
3+1
2+2
8
LUCULLUS
4
4
4

 

 

-
7
L
U
C
U
L
L
U
S
-
-
-
-
-
-
-
-
-
-
-
-
-
--
-
-
-
-
-
1
+
=
10
1+0
=
1
=
1
=
1
-
-
-
--
-
-
-
-
-
19
+
=
19
1+9
=
10
1+0
1
=
1
-
7
L
U
C
U
L
L
U
S
-
-
-
-
-
-
-
-
-
-
-
-
3
3
3
3
3
3
3
-
+
=
21
2+1
=
3
=
3
=
3
-
-
12
21
3
21
12
12
21
-
+
=
30
3+0
=
3
=
3
=
3
-
7
L
U
C
U
L
L
U
S
-
-
-
-
-
-
-
-
-
-
-
-
12
21
3
21
12
12
21
19
+
=
121
1+2+1
=
4
=
4
=
4
-
-
3
3
3
3
3
3
3
1
+
=
22
2+2
=
4
=
4
=
4
-
7
L
U
C
U
L
L
U
S
-
-
-
-
-
-
-
-
-
-
-
-
-
--
-
--
-
-
--
1
-
-
1
occurs
x
1
=
1
=
1
2
-
-
--
-
--
-
-
--
-
-
-
2
TWO
2
-
-
-
-
-
-
-
3
3
3
3
3
3
3
-
-
-
3
occurs
x
7
=
21
2+1
3
4
-
-
--
-
--
-
-
--
-
-
-
4
FOUR
4
-
-
-
-
-
5
-
-
--
-
--
-
-
--
-
-
-
5
FIVE
5
-
-
-
-
-
6
-
-
--
-
--
-
-
--
-
-
-
6
SIX
6
-
-
-
-
-
7
-
-
--
-
--
-
-
--
-
-
-
7
SEVEN
7
-
-
-
-
-
8
-
-
--
-
--
-
-
--
-
-
-
8
EIGHT
8
-
-
-
-
-
9
-
-
--
-
--
-
-
--
-
-
-
9
NINE
9
-
-
-
-
-
41
8
L
U
C
U
L
L
U
S
-
-
4
-
-
8
-
22
-
4
4+1
-
3
3
3
3
3
3
3
-
-
-
-
-
-
-
-
2+2
-
-
5
8
L
U
C
U
L
L
U
S
-
-
4
-
-
8
-
4
-
4
-
-
3
3
3
3
3
3
3
1
-
-
-
-
-
-
-
-
-
-
5
8
L
U
C
U
L
L
U
S
-
-
4
-
-
8
-
4
-
4

 

 

7
L
U
C
U
L
L
U
S
-
-
-
-
-
-
-
-
-
-
-
-
--
-
-
-
-
-
1
+
=
10
1+0
=
1
=
1
=
1
-
-
--
-
-
-
-
-
19
+
=
19
1+9
=
10
1+0
1
=
1
7
L
U
C
U
L
L
U
S
-
-
-
-
-
-
-
-
-
-
-
3
3
3
3
3
3
3
-
+
=
21
2+1
=
3
=
3
=
3
-
12
21
3
21
12
12
21
-
+
=
30
3+0
=
3
=
3
=
3
7
L
U
C
U
L
L
U
S
-
-
-
-
-
-
-
-
-
-
-
12
21
3
21
12
12
21
19
+
=
121
1+2+1
=
4
=
4
=
4
-
3
3
3
3
3
3
3
1
+
=
22
2+2
=
4
=
4
=
4
7
L
U
C
U
L
L
U
S
-
-
-
-
-
-
-
-
-
-
-
-
--
-
--
-
-
--
1
-
-
1
occurs
x
1
=
1
=
1
-
3
3
3
3
3
3
3
-
-
-
3
occurs
x
7
=
21
2+1
3
8
L
U
C
U
L
L
U
S
-
-
4
-
-
8
-
22
-
4
-
3
3
3
3
3
3
3
-
-
-
-
-
-
-
-
2+2
-
-
8
L
U
C
U
L
L
U
S
-
-
4
-
-
8
-
4
-
4
-
3
3
3
3
3
3
3
1
-
-
-
-
-
-
-
-
-
-
8
L
U
C
U
L
L
U
S
-
-
4
-
-
8
-
4
-
4

 

 

Daily Mail, Thursday, November 17, 2011

ANSWERS TO CORRESPONDENTS

Compiled by Charles Legge

Page 70

QUESTION The W. B Yeats poem Mad As The Mists and snow name checks Homer, Horace, Cicero and and Tully ('and here is Tully's open page') Who was Tully?


TULLY and Cicero are one and the same person - the Roman statesman and philosopher whose full latin name was Marcus Tullius Cicero (106-43 BC).

Male Roman citizens typically had three names, their praenomen (given name), their nomen (clan name) and their' cognomen (nickname or family name), but in English classical Latin authors are generally known by a single name, often Anglicised. These sometimes derive from the nomen (for example, Horace's full name was Quintus Horatius Flaccus) and sometimes from the cognomen (as with Gaius Julius Caesar).

In Cicero's case, he's today known by his cognomen, but has sometimes in the past been called Tully from his nomen.

Despite coming from humble stock, Cicero's skill as an orator saw him rise to the supreme position of state in the Republic, the consulship, in 63BC and he played a key role in the defeat of the Catiline conspiracy.

As a result of the rise to power of Julius Caesar, he took a lesser role in his later years, devoting himself to literature Instead.

After Caesar's assassination in 44BC, Cicero returned to politics and took a key role in the senatorial opposition to Mark Antony.

He was murdered on Antony's orders when he took power. Cicero's surviving letters form a key source for the history of the fall of the Roman Republic and its replacement by the rule of the emperors.

David Bradbury, London.

 

-
TULLIUS
-
-
-
1
T
20
2
2
1
U
21
3
3
1
L
12
3
3
1
L
12
3
3
1
I
9
9
9
1
U
21
3
3
1
S
19
10
1
7
TULLIUS
114
33
24
-
-
1+1+4
3+3
2+4
7
TULLIUS
6
6
6

 

 

-
-
-
-
-
TULLIUS
-
-
-
-
1
2
3
4
5
6
7
8
9
T
=
2
2
1
T
20
2
2
-
-
2
-
-
-
-
-
-
-
U
=
3
-
1
U
21
3
3
-
-
-
3
-
-
-
-
-
-
L
=
3
-
1
L
12
3
3
-
-
-
3
-
-
-
-
-
-
L
=
3
-
1
L
12
3
3
-
-
-
3
-
-
-
-
-
-
I
=
9
9
1
I
9
9
9
-
-
-
-
-
-
-
-
-
-
U
=
3
-
1
U
21
3
3
-
-
-
3
-
-
-
-
-
-
S
=
1
1
1
S
19
10
1
-
1
-
-
-
-
-
-
-
-
-
-
24
12
7
TULLIUS
114
33
24
-
1
2
12
-
-
-
-
-
9
-
-
2+4
1+2
-
-
1+1+4
3+3
2+4
-
-
-
1+2
-
-
-
-
-
-
-
-
6
3
7
TULLIUS
6
6
6
-
1
2
3
-
-
-
-
-
9

 

 

-
7
T
U
L
L
I
U
S
-
-
-
-
-
-
-
-
-
-
-
-
-
--
-
-
9
-
1
+
=
10
1+0
=
1
=
1
=
1
-
-
-
--
-
-
9
-
19
+
=
28
2+8
=
10
1+0
1
=
1
-
7
T
U
L
L
I
U
S
-
-
-
-
-
-
-
-
-
-
-
-
2
3
3
3
-
3
-
+
=
14
1+4
=
5
=
5
=
5
-
-
20
21
12
12
-
21
-
+
=
86
8+6
=
14
1+4
5
=
5
-
7
T
U
L
L
I
U
S
-
-
-
-
-
-
-
-
-
-
-
-
20
21
12
12
9
21
19
+
=
114
1+1+4
=
6
=
6
=
6
-
-
2
3
3
3
9
3
1
+
=
33
3+3
=
6
=
6
=
6
-
7
T
U
L
L
I
U
S
-
-
-
-
-
-
-
-
-
-
-
-
-
--
-
-
-
--
1
-
-
1
occurs
x
1
=
1
=
1
-
-
2
--
-
-
-
--
-
-
-
2
occurs
x
1
=
2
=
2
-
-
-
3
3
3
-
3
-
-
-
3
occurs
x
4
=
12
1+2
3
4
-
-
--
-
-
-
--
-
-
-
4
FOUR
4
-
-
-
-
-
5
-
-
--
-
-
-
--
-
-
-
5
FIVE
5
-
-
-
-
-
6
-
-
--
-
-
-
--
-
-
-
6
SIX
6
-
-
-
-
-
7
-
-
--
-
-
-
--
-
-
-
7
SEVEN
7
-
-
-
-
-
8
-
-
--
-
-
-
--
-
-
-
8
EIGHT
8
-
-
-
-
-
-
-
-
--
-
-
9
--
-
-
-
9
occurs
x
1
=
9
=
9
30
7
T
U
L
L
I
U
S
-
-
15
-
-
7
-
24
-
15
3+0
-
-
3
3
3
-
3
-
-
-
1+5
-
-
-
-
2+4
-
1+5
3
7
T
U
L
L
I
U
S
-
-
6
-
-
7
-
6
-
6
-
-
2
3
3
3
9
3
1
-
-
-
-
-
-
-
-
-
-
3
7
T
U
L
L
I
U
S
-
-
6
-
-
7
-
6
-
6

 

 

7
T
U
L
L
I
U
S
-
-
-
-
-
-
-
-
-
-
-
-
--
-
-
9
-
1
+
=
10
1+0
=
1
=
1
=
1
-
-
--
-
-
9
-
19
+
=
28
2+8
=
10
1+0
1
=
1
7
T
U
L
L
I
U
S
-
-
-
-
-
-
-
-
-
-
-
2
3
3
3
-
3
-
+
=
14
1+4
=
5
=
5
=
5
-
20
21
12
12
-
21
-
+
=
86
8+6
=
14
1+4
5
=
5
7
T
U
L
L
I
U
S
-
-
-
-
-
-
-
-
-
-
-
20
21
12
12
9
21
19
+
=
114
1+1+4
=
6
=
6
=
6
-
2
3
3
3
9
3
1
+
=
33
3+3
=
6
=
6
=
6
7
T
U
L
L
I
U
S
-
-
-
-
-
-
-
-
-
-
-
-
--
-
-
-
--
1
-
-
1
occurs
x
1
=
1
=
1
-
2
--
-
-
-
--
-
-
-
2
occurs
x
1
=
2
=
2
-
-
3
3
3
-
3
-
-
-
3
occurs
x
4
=
12
1+2
3
-
-
--
-
-
9
--
-
-
-
9
occurs
x
1
=
9
=
9
7
T
U
L
L
I
U
S
-
-
15
-
-
7
-
24
-
15
-
-
3
3
3
-
3
-
-
-
1+5
-
-
-
-
2+4
-
1+5
7
T
U
L
L
I
U
S
-
-
6
-
-
7
-
6
-
6
-
2
3
3
3
9
3
1
-
-
-
-
-
-
-
-
-
-
7
T
U
L
L
I
U
S
-
-
6
-
-
7
-
6
-
6

 

 

7
T
U
L
L
I
U
S
-
-
-
-
-
-
-
-
-
-
-
20
21
12
12
9
21
19
+
=
114
1+1+4
=
6
=
6
=
6
-
2
3
3
3
9
3
1
+
=
33
3+3
=
6
=
6
=
6
7
T
U
L
L
I
U
S
-
-
-
-
-
-
-
-
-
-
-
-
--
-
-
-
--
1
-
-
1
occurs
x
1
=
1
=
1
-
2
--
-
-
-
--
-
-
-
2
occurs
x
1
=
2
=
2
-
-
3
3
3
-
3
-
-
-
3
occurs
x
4
=
12
1+2
3
-
-
--
-
-
9
--
-
-
-
9
occurs
x
1
=
9
=
9
7
T
U
L
L
I
U
S
-
-
15
-
-
7
-
24
-
15
-
-
3
3
3
-
3
-
-
-
1+5
-
-
-
-
2+4
-
1+5
7
T
U
L
L
I
U
S
-
-
6
-
-
7
-
6
-
6
-
2
3
3
3
9
3
1
-
-
-
-
-
-
-
-
-
-
7
T
U
L
L
I
U
S
-
-
6
-
-
7
-
6
-
6

 

 

THE ART OF MEMORY

FRANCIS A. YATES 1966

THREE LATIN SOURCES FOR THE CLASSICAL ART OF MEMORY

Page 21

"The Ad Herennium was a well known and much used text in the Middle Ages when it had an immense prestige because it was thought to be by Cicero. It was therefore believed that the precepts for the artificial memory which it expounded had been drawn up by Tullius himself"

Page 32

"Though the medieval tradition which assigned the authorship of Ad Herennium to 'Tullius' was wrong in fact, it was not wrong in its inference that the art of memory was practised and recommended by Tullius."

Page 282

"By the THE LADDER OF MINERVA we rise from the first to the last..."

 

 

A
=
1
-
2
AD
5
5
5
H
=
8
-
9
HERENNIUM
107
53
8
S
-
9
4
11
First Total
112
58
13
-
S
-
-
1+1
Add to Reduce
1+1+2
5+8
1+3
S
-
9
4
2
Second Total
4
13
4
-
-
-
-
-
Reduce to Deduce
1+2
1+3
-
-
-
9
-
2
Essence of Number
4
4
4

 

 

-
11
A
D
-
H
E
R
E
N
N
I
U
M
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
8
-
-
-
5
5
9
-
-
+
=
27
2+7
=
9
=
9
=
9
-
-
-
-
-
8
-
-
-
14
14
9
-
-
+
=
45
4+5
=
9
=
9
=
9
-
11
A
D
-
H
E
R
E
N
N
I
U
M
-
-
-
-
-
-
-
-
-
-
-
-
1
4
-
-
5
9
5
-
-
-
3
4
+
=
31
3+1
=
4
=
4
=
4
-
-
1
4
-
-
5
18
5
-
-
-
21
13
+
=
67
6+7
=
13
1+3
4
=
4
-
11
A
D
-
H
E
R
E
N
N
I
U
M
-
-
-
-
-
-
-
-
-
-
-
-
1
4
-
8
5
18
5
14
14
9
21
13
+
=
112
1+1+2
=
4
=
4
=
4
-
-
1
4
-
8
5
9
5
5
5
9
3
4
+
=
58
5+8
=
13
1+3
4
=
4
-
11
A
D
-
H
E
R
E
N
N
I
U
M
-T
-
-
-
-
-
-
-
-
-
-
-
1
-
-
-
-
-
-
-
-
-
-
-
-
-
1
occurs
x
1
=
1
=
1
2
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
2
TWO
2
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
3
-
-
-
3
occurs
x
1
=
3
=
3
-
-
-
4
-
-
-
-
-
-
-
-
-
4
-
-
4
occurs
x
2
=
8
=
8
-
-
-
-
-
-
5
-
5
5
5
-
-
-
-
-
5
occurs
x
4
=
20
2+0
2
6
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
6
SIX
6
-
-
-
-
-
7
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
7
SEVEN
7
-
-
-
-
-
-
-
-
-
-
8
-
-
-
-
-
-
-
-
-
-
8
occurs
x
1
=
8
=
8
-
-
-
-
-
-
-
9
-
-
-
9
-
-
-
-
9
occurs
x
1
=
18
1+8
9
15
11
A
D
-
H
E
R
E
N
N
I
U
M
-
-
30
-
1
11
-
58
-
31
1+5
1+1
-
-
-
-
-
9
-
-
-
9
-
-
-
-
3+0
-
-
1+1
-
5+8
-
3+1
6
2
A
D
-
H
E
R
E
N
N
I
U
M
-
-
3
-
1
2
-
13
-
4
-
-
1
4
-
8
5
9
5
5
5
9
3
4
-
-
-
-
-
-
-
1+3
-
-
6
2
A
D
-
H
E
R
E
N
N
I
U
M
-
-
3
-
-
2
-
4
-
4

 

 

1
A
D
-
H
E
R
E
N
N
I
U
M
-
-
-
-
-
-
-
-
-
-
-
-
-
-
8
-
-
-
5
5
9
-
-
+
=
27
2+7
=
9
=
9
=
9
-
-
-
-
8
-
-
-
14
14
9
-
-
+
=
45
4+5
=
9
=
9
=
9
11
A
D
-
H
E
R
E
N
N
I
U
M
-
-
-
-
-
-
-
-
-
-
-
1
4
-
-
5
9
5
-
-
-
3
4
+
=
31
3+1
=
4
=
4
=
4
-
1
4
-
-
5
18
5
-
-
-
21
13
+
=
67
6+7
=
13
1+3
4
=
4
11
A
D
-
H
E
R
E
N
N
I
U
M
-
-
-
-
-
-
-
-
-
-
-
1
4
-
8
5
18
5
14
14
9
21
13
+
=
112
1+1+2
=
4
=
4
=
4
-
1
4
-
8
5
9
5
5
5
9
3
4
+
=
58
5+8
=
13
1+3
4
=
4
11
A
D
-
H
E
R
E
N
N
I
U
M
-T
-
-
-
-
-
-
-
-
-
-
1
-
-
-
-
-
-
-
-
-
-
-
-
-
1
occurs
x
1
=
1
=
1
-
-
-
-
-
-
-
-
-
-
-
3
-
-
-
3
occurs
x
1
=
3
=
3
-
-
4
-
-
-
-
-
-
-
-
-
4
-
-
4
occurs
x
2
=
8
=
8
-
-
-
-
-
5
-
5
5
5
-
-
-
-
-
5
occurs
x
4
=
20
2+0
2
-
-
-
-
8
-
-
-
-
-
-
-
-
-
-
8
occurs
x
1
=
8
=
8
-
-
-
-
-
-
9
-
-
-
9
-
-
-
-
9
occurs
x
1
=
18
1+8
9
11
A
D
-
H
E
R
E
N
N
I
U
M
-
-
30
-
1
11
-
58
-
31
1+1
-
-
-
-
-
9
-
-
-
9
-
-
-
-
3+0
-
-
1+1
-
5+8
-
3+1
2
A
D
-
H
E
R
E
N
N
I
U
M
-
-
3
-
1
2
-
13
-
4
-
1
4
-
8
5
9
5
5
5
9
3
4
-
-
-
-
-
-
-
1+3
-
-
2
A
D
-
H
E
R
E
N
N
I
U
M
-
-
3
-
-
2
-
4
-
4

 

 

1
A
D
H
E
R
E
N
N
I
U
M
-
-
-
-
-
-
-
-
-
-
-
-
-
8
-
-
-
5
5
9
-
-
+
=
27
2+7
=
9
=
9
=
9
-
-
-
8
-
-
-
14
14
9
-
-
+
=
45
4+5
=
9
=
9
=
9
11
A
D
H
E
R
E
N
N
I
U
M
-
-
-
-
-
-
-
-
-
-
-
1
4
-
5
9
5
-
-
-
3
4
+
=
31
3+1
=
4
=
4
=
4
-
1
4
-
5
18
5
-
-
-
21
13
+
=
67
6+7
=
13
1+3
4
=
4
11
A
D
H
E
R
E
N
N
I
U
M
-
-
-
-
-
-
-
-
-
-
-
1
4
8
5
18
5
14
14
9
21
13
+
=
112
1+1+2
=
4
=
4
=
4
-
1
4
8
5
9
5
5
5
9
3
4
+
=
58
5+8
=
13
1+3
4
=
4
11
A
D
H
E
R
E
N
N
I
U
M
-T
-
-
-
-
-
-
-
-
-
-
1
-
-
-
-
-
-
-
-
-
-
-
-
1
occurs
x
1
=
1
=
1
-
-
-
-
-
-
-
-
-
-
3
-
-
-
3
occurs
x
1
=
3
=
3
-
-
4
-
-
-
-
-
-
-
-
4
-
-
4
occurs
x
2
=
8
=
8
-
-
-
-
5
-
5
5
5
-
-
-
-
-
5
occurs
x
4
=
20
2+0
2
-
-
-
8
-
-
-
-
-
-
-
-
-
-
8
occurs
x
1
=
8
=
8
-
-
-
-
-
9
-
-
-
9
-
-
-
-
9
occurs
x
1
=
18
1+8
9
11
A
D
H
E
R
E
N
N
I
U
M
-
-
30
-
1
11
-
58
-
31
1+1
-
-
-
-
9
-
-
-
9
-
-
-
-
3+0
-
-
1+1
-
5+8
-
3+1
2
A
D
H
E
R
E
N
N
I
U
M
-
-
3
-
1
2
-
13
-
4
-
1
4
8
5
9
5
5
5
9
3
4
-
-
-
-
-
-
-
1+3
-
-
2
A
D
H
E
R
E
N
N
I
U
M
-
-
3
-
-
2
-
4
-
4

 

 

-
-
-
-
-
AD HERENNIUM
-
-
-
A
=
1
-
2
A+D
5
5
5
-
-
-
-
9
HERENNIUM
-
-
-
-
S
-
S
S
H
8
8
8
S
-
S
-
-
E
5
5
5
-
S
-
S
S
R
18
9
9
S
-
S
-
-
E
5
5
5
-
S
-
S
S
N
14
5
5
-
S
-
S
S
N
14
5
5
-
S
-
S
S
I
9
9
9
S
-
S
-
-
U
21
3
3
S
-
S
-
-
M
13
4
4
S
S
-
S
-
AD HERENNIUM
-
-
-
H
=
8
4
11
First Total
112
58
58
-
S
-
-
1+1
Add to Reduce
1+1+2
5+8
5+8
S
-
9
4
2
Second Total
4
13
13
-
-
-
-
-
Reduce to Deduce
1+2
1+3
1+3
-
-
9
-
2
Essence of Number
4
4
4

 

 

THE ART OF MEMORY

FRANCIS A. YATES 1966

THREE LATIN SOURCES FOR THE CLASSICAL ART OF MEMORY

Page 175

CHAPTER 8

LULLISM AS AN ART OF MEMORY

THOUGH we have now reached the Renaissance, with Camillo, we have to retrace our steps to the Middle Ages during this chapter. For there was another kind of art of memory which began in the Middle Ages, which continued into the Renaissance and beyond, and which it was the aim of many in the Renaissance to combine with the classical art in some new synthesis whereby memory should reach still further heights of insight and of power. This other art of memory was the Art of Ramon Lull.

Lullism and its history is a most difficult subject and one for the exploration of which the full materials have not yet been assembled; The enormous number of Lull's own writings, some of them still unpublished, the vast Lullist literature written by his followers, the extreme complexity of Lullism, make it impossible as yet to reach very definite conclusions about what is, undoubtedly.ra strand of major importance in the European tradition. And what I have to do now is to write one not very long chapter giving some idea of what the Art of Ramon Lullwas like, of why it was an art of memory, of how it differs from the classical art of memory, and of how Lullism became absorbed at the Renaissance into Renaissance forms of the classical art.

Obviously- I am attempting the impossible, yet the impossible must be attempted because it is essential for the later part of this book that there should be some sketch at this stage of Lullism itself. The chapter is based on my own two articles on the art of Ramon Lull:' it is orientated towards a comparison of Lullism as an art of memory with. the classical art; and it is not concerned solely with 'genuine' Lullism but also with the Renaissance interpretation of Lullism, for it is this. which is important for the next stages of our history.

 

ART AT R AT ART

 

 

8
SIG1LLUS
108
36
9
10
SIGILLORUM
135
54
9
18
-
243
90
18
1+8
-
2+4+3
9+0
1+8
9
-
9
9
9

 

 

-
18
S
I
G
I
L
L
U
S
-
S
I
G
I
L
L
O
R
U
M
-
-
-
-
-
-
-
-
-
-
-
-
1
9
-
9
-
-
-
1
-
1
9
-
9
-
-
6
-
-
-
+
=
45
4+5
=
9
=
9
=
9
-
-
19
9
-
9
-
-
-
19
-
19
9
-
9
-
-
15
-
-
-
+
=
108
1+0+8
=
9
=
9
=
9
-
18
S
I
G
I
L
L
U
S
-
S
I
G
I
L
L
O
R
U
M
-
-
-
-
-
-
-
-
-
-
-
-
-
-
7
-
3
3
3
-
-
-
-
7
-
3
3
-
9
3
4
+
=
45
4+5
=
9
=
9
=
9
-
-
-
-
7
-
12
12
21
-
-
-
-
7
-
12
12
-
18
21
13
+
=
135
1+3+5
=
9
=
9
=
9
-
18
S
I
G
I
L
L
U
S
-
S
I
G
I
L
L
O
R
U
M
-
-
-
-
-
-
-
-
-
-
-
-
19
9
7
9
12
12
21
19
-
19
9
7
9
12
12
15
18
21
13
+
=
243
2+4+3
=
9
=
9
=
9
-
-
1
9
7
9
3
3
3
1
-
1
9
7
9
3
3
6
9
3
4
+
=
90
9+0
=
9
=
9
=
9
-
18
S
I
G
I
L
L
U
S
-
S
I
G
I
L
L
O
R
U
M
-
-
-
-
-
-
-
-
-
-
-
-
1
-
-
-
-
-
-
1
-
1
-
-
-
-
-
-
-
-
-
-
-
1
occurs
x
3
=
3
=
3
2
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
2
TWO
2
-
-
-
-
-
-
-
-
-
-
-
3
3
3
-
-
-
-
-
-
3
3
-
-
3
-
-
-
3
occurs
x
6
=
18
1+8
9
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
4
-
-
4
occurs
x
1
=
4
=
4
5
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
5
FIVE
5
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
6
-
-
-
-
-
6
occurs
x
1
=
6
=
6
-
-
-
-
7
-
-
-
-
-
-
-
-
7
-
-
-
-
-
-
-
-
-
9
occurs
x
2
=
14
1+4
5
8
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
8
EIGHT
8
-
-
-
-
-
-
-
-
9
-
9
-
-
-
-
-
-
9
-
9
-
-
-
9
-
-
-
-
9
occurs
x
5
=
45
4+5
9
15
18
S
I
G
I
L
L
U
S
-
S
I
G
I
L
L
O
R
U
M
-
-
30
-
-
18
-
90
-
36
1+5
1+8
-
-
-
-
3
3
3
-
-
-
-
-
-
3
3
-
-
3
-
-
-
3+0
-
-
1+8
-
9+0
-
3+6
6
9
S
I
G
I
L
L
U
S
-
S
I
G
I
L
L
O
R
U
M
-
-
3
-
-
9
-
9
-
9
-
-
1
9
7
9
3
3
3
1
-
1
9
7
9
3
3
6
9
3
4
-
-
-
-
-
-
-
-
-
-
6
9
S
I
G
I
L
L
U
S
-
S
I
G
I
L
L
O
R
U
M
-
-
3
-
-
9
-
9
-
9

 

 

18
S
I
G
I
L
L
U
S
-
S
I
G
I
L
L
O
R
U
M
-
-
-
-
-
-
-
-
-
-
-
1
9
-
9
-
-
-
1
-
1
9
-
9
-
-
6
-
-
-
+
=
45
4+5
=
9
=
9
=
9
-
19
9
-
9
-
-
-
19
-
19
9
-
9
-
-
15
-
-
-
+
=
108
1+0+8
=
9
=
9
=
9
18
S
I
G
I
L
L
U
S
-
S
I
G
I
L
L
O
R
U
M
-
-
-
-
-
-
-
-
-
-
-
-
-
7
-
3
3
3
-
-
-
-
7
-
3
3
-
9
3
4
+
=
45
4+5
=
9
=
9
=
9
-
-
-
7
-
12
12
21
-
-
-
-
7
-
12
12
-
18
21
13
+
=
135
1+3+5
=
9
=
9
=
9
18
S
I
G
I
L
L
U
S
-
S
I
G
I
L
L
O
R
U
M
-
-
-
-
-
-
-
-
-
-
-
19
9
7
9
12
12
21
19
-
19
9
7
9
12
12
15
18
21
13
+
=
243
2+4+3
=
9
=
9
=
9
-
1
9
7
9
3
3
3
1
-
1
9
7
9
3
3
6
9
3
4
+
=
90
9+0
=
9
=
9
=
9
18
S
I
G
I
L
L
U
S
-
S
I
G
I
L
L
O
R
U
M
-
-
-
-
-
-
-
-
-
-
-
1
-
-
-
-
-
--
1
-
1
-
-
-
-
-
-
-
--
-
-
-
1
occurs
x
3
=
3
=
3
-
-
-
-
-
3
3
3
-
-
-
-
-
-
3
3
-
-
3
-
-
-
3
occurs
x
6
=
18
1+8
9
-
-
-
-
-
-
-
--
-
-
-
-
-
-
-
-
-
-
--
4
-
-
4
occurs
x
1
=
4
=
4
-
-
-
-
-
-
-
--
-
-
-
-
-
-
-
-
6
-
--
-
-
-
6
occurs
x
1
=
6
=
6
-
-
-
7
-
-
-
--
-
-
-
-
7
-
-
-
-
-
--
-
-
-
9
occurs
x
2
=
14
1+4
5
-
-
9
-
9
-
-
--
-
-
-
9
-
9
-
-
-
9
--
-
-
-
9
occurs
x
5
=
45
4+5
9
18
S
I
G
I
L
L
U
S
-
S
I
G
I
L
L
O
R
U
M
-
-
30
-
-
18
-
90
-
36
1+8
-
-
-
-
3
3
3
-
-
-
-
-
-
3
3
-
-
3
-
-
-
3+0
-
-
1+8
-
9+0
-
3+6
9
S
I
G
I
L
L
U
S
-
S
I
G
I
L
L
O
R
U
M
-
-
3
-
-
9
-
9
-
9
-
1
9
7
9
3
3
3
1
-
1
9
7
9
3
3
6
9
3
4
-
-
-
-
-
-
-
-
-
-
9
S
I
G
I
L
L
U
S
-
S
I
G
I
L
L
O
R
U
M
-
-
3
-
-
9
-
9
-
9

 

 

18
S
I
G
I
L
L
U
S
S
I
G
I
L
L
O
R
U
M
-
-
-
-
-
-
-
-
-
-
-
1
9
-
9
-
-
-
1
1
9
-
9
-
-
6
-
-
-
+
=
45
4+5
=
9
=
9
=
9
-
19
9
-
9
-
-
-
19
19
9
-
9
-
-
15
-
-
-
+
=
108
1+0+8
=
9
=
9
=
9
18
S
I
G
I
L
L
U
S
S
I
G
I
L
L
O
R
U
M
-
-
-
-
-
-
-
-
-
-
-
-
-
7
-
3
3
3
-
-
-
7
-
3
3
-
9
3
4
+
=
45
4+5
=
9
=
9
=
9
-
-
-
7
-
12
12
21
-
-
-
7
-
12
12
-
18
21
13
+
=
135
1+3+5
=
9
=
9
=
9
18
S
I
G
I
L
L
U
S
S
I
G
I
L
L
O
R
U
M
-
-
-
-
-
-
-
-
-
-
-
19
9
7
9
12
12
21
19
19
9
7
9
12
12
15
18
21
13
+
=
243
2+4+3
=
9
=
9
=
9
-
1
9
7
9
3
3
3
1
1
9
7
9
3
3
6
9
3
4
+
=
90
9+0
=
9
=
9
=
9
18
S
I
G
I
L
L
U
S
S
I
G
I
L
L
O
R
U
M
-
-
-
-
-
-
-
-
-
-
-
1
-
-
-
-
-
--
1
1
-
-
-
-
-
-
-
--
-
-
-
1
occurs
x
3
=
3
=
3
-
-
-
-
-
3
3
3
-
-
-
-
-
3
3
-
-
3
-
-
-
3
occurs
x
6
=
18
1+8
9
-
-
-
-
-
-
-
--
-
-
-
-
-
-
-
-
-
--
4
-
-
4
occurs
x
1
=
4
=
4
-
-
-
-
-
-
-
--
-
-
-
-
-
-
-
6
-
--
-
-
-
6
occurs
x
1
=
6
=
6
-
-
-
7
-
-
-
--
-
-
-
7
-
-
-
-
-
--
-
-
-
9
occurs
x
2
=
14
1+4
5
-
-
9
-
9
-
-
--
-
-
9
-
9
-
-
-
9
--
-
-
-
9
occurs
x
5
=
45
4+5
9
18
S
I
G
I
L
L
U
S
S
I
G
I
L
L
O
R
U
M
-
-
30
-
-
18
-
90
-
36
1+8
-
-
-
-
3
3
3
-
-
-
-
-
3
3
-
-
3
-
-
-
3+0
-
-
1+8
-
9+0
-
3+6
9
S
I
G
I
L
L
U
S
S
I
G
I
L
L
O
R
U
M
-
-
3
-
-
9
-
9
-
9
-
1
9
7
9
3
3
3
1
1
9
7
9
3
3
6
9
3
4
-
-
-
-
-
-
-
-
-
-
9
S
I
G
I
L
L
U
S
S
I
G
I
L
L
O
R
U
M
-
-
3
-
-
9
-
9
-
9

 

 

IN

MEMORIAM

"SUFFER LITTLE CHILDREN TO COME UNTO ME"

 

 

Children of Llullaillaco, sacrificed by the Incas 500 years ago.

It is believed the Children of Llullaillaco, as they have come to be known, were sacrificed during a ceremony thanking the Inca gods for the annual corn ... www.democraticunderground.com/discuss/duboard.

http://news.bbc.co.uk/1/hi/world/americas/6983300.stm

Mummified Inca maiden wows crowds

"A mummy of an Inca girl, described as "perfect" by the archaeologists who found her in 1999, has gone on display for the first time in Argentina .

Hundreds of people crowded into a museum in the north-western city of Salta to see "la Doncella", the Maiden.

The remains of the girl, who was 15 when she died, were found in an icy pit on top of a volcano in the Andes, along with a younger boy and girl. Researchers believe they were sacrificed by the Incas 500 years ago. The three were discovered at a height of 6,700m (22,000ft) on Mount Llullaillaco, a volcano in north-west Argentina on the border with Chile. At the time, the archaeologist leading the team, Dr Johan Reinhard, said they appeared "the best preserved of any mummy I've seen". It is believed the Children of Llullaillaco, as they have come to be known, were sacrificed during a ceremony thanking the Inca gods for the annual corn harvest.

'Great mistake'

The mummy of la Doncella is on display in a chamber that is filled with cold air that recreates the sub-freezing conditions in which she was found. Visitors told Argentina media they were impressed at the mummy's state of conservation. "I'm amazed," one woman said. "You just expect her at any moment to get up and start talking." But the exhibition has angered several indigenous groups who campaigned to stop the mummy from going on display.

Miguel Suarez from the Calchaquies valley tribes in and around Salta told the Associated Press news agency that the exhibit was "a great mistake", adding that he hoped visitors would show respect for the dead."

 

 

-
LLULLAILLACO

-

-
-
3
L+L+U

45

9
9
2
L+L+A

25

7
7
1
I

9

9
9
3
L+L+A

25

7
7
2
C+O

18

9
9
12
LLULLAILLACO

122

41

41
1+2
-
1+2+2
4+1
4+1
3
LLULLAILLACO

5

5

5
-
-
-
-
-
3
LLULLAILLACO

5

5

5

 

 

-
-
-
-
-
-
-
-
LLULLAILLACO

-

-
-
L
3
-
-
-
-
L
1
L

12

3
3
L
3
-
-
-
-
L
1
L

12

3
3
U
3
-
-
-
-
U
1
U

21

3
3
L
3
-
-
-
-
L
1
L

12

3
3
L
3
-
-
-
-
L
1
L

12

3
3
-
-
A
-
-
-
A
1
A

1

1
1
-
-
-
I
-
-
I
1
I

9

9
9
L
3
-
-
-
-
L
1
L

12

3
3
L
3
-
-
-
-
L
1
L

12

3
3
-
-
A
-
-
-
A
1
A

1

1
1
C
3
-
-
-
-
C
1
C

3

3
3
-
-
-
-
O
-
O
1
O

15

6
6
-
24
-
-
-
-
-
12
LLULLAILLACO

122

41

41
-
2+4
-
-
-
-
-
1+2
-
1+2+2
4+1
4+1
-
6
-
-
-
-
-
3
LLULLAILLACO

5

5

5
-
-
-
-
-
-
-
-
-
-
-
-
-
1
-
-
-
-
-
3
LLULLAILLACO

5

5

5

 

 

-
12
L
L
U
L
L
A
I
L
L
A
C
O
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
9
-
-
-
-
6
+
=
15
1+5
=
6
=
6
=
6
-
-
-
-
-
-
-
-
9
-
-
-
-
15
+
=
24
2+4
=
6
=
6
=
6
-
12
L
L
U
L
L
A
I
L
L
A
C
O
-
-
-
-
-
-
-
-
-
-
-
-
3
3
3
3
3
1
-
3
3
1
3
-
+
=
26
2+6
=
8
=
8
=
8
-
-
12
12
21
12
12
1
-
12
12
1
3
-
+
=
98
9+8
=
17
1+7
8
=
8
-
12
L
L
U
L
L
A
I
L
L
A
C
O
-
-
-
-
-
-
-
-
-
-
-
-
12
12
21
12
12
1
9
12
12
1
3
15
+
=
122
1+2+2
=
5
=
5
=
5
-
-
3
3
3
3
3
1
9
3
3
1
3
6
+
=
41
4+1
=
5
=
5
=
5
-
12
L
L
U
L
L
A
I
L
L
A
C
O
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
1
-
-
-
1
-
-
-
-
1
occurs
x
2
=
2
=
2
2
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
2
TWO
2
-
-
-
-
-
-
-
3
3
3
3
3
-
-
3
3
-
3
--
-
-
3
occurs
x
8
=
24
2+4
6
4
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
4
FOUR
4
-
-
-
-
-
5
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
5
FIVE
8
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
6
-
-
6
occurs
x
1
=
6
=
6
7
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
7
SEVEN
7
-
-
-
-
-
8
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
8
EIGHT
8
-
-
-
-
-
-
-
-
-
-
-
-
-
9
-
-
-
-
-
-
-
9
occurs
x
1
=
9
=
9
26
12
L
L
U
L
L
A
I
L
L
A
C
O
-
-
19
-
-
12
-
41
-
23
2+6
1+2
3
3
3
3
3
-
-
3
3
-
3
--
-
-
1+9
-
-
1+2
-
4+1
-
2+3
8
3
L
L
U
L
L
A
I
L
L
A
C
O
-
-
10
-
-
3
-
5
-
5
-
-
3
3
3
3
3
1
9
3
3
1
3
6
-
-
1+0
-
-
-
-
-
-
-
8
3
L
L
U
L
L
A
I
L
L
A
C
O
-
-
1
-
-
3
-
5
-
5

 

 

12
L
L
U
L
L
A
I
L
L
A
C
O
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
9
-
-
-
-
6
+
=
15
1+5
=
6
=
6
=
6
-
-
-
-
-
-
-
9
-
-
-
-
15
+
=
24
2+4
=
6
=
6
=
6
12
L
L
U
L
L
A
I
L
L
A
C
O
-
-
-
-
-
-
-
-
-
-
-
3
3
3
3
3
1
-
3
3
1
3
-
+
=
26
2+6
=
8
=
8
=
8
-
12
12
21
12
12
1
-
12
12
1
3
-
+
=
98
9+8
=
17
1+7
8
=
8
12
L
L
U
L
L
A
I
L
L
A
C
O
-
-
-
-
-
-
-
-
-
-
-
12
12
21
12
12
1
9
12
12
1
3
15
+
=
122
1+2+2
=
5
=
5
=
5
-
3
3
3
3
3
1
9
3
3
1
3
6
+
=
41
4+1
=
5
=
5
=
5
12
L
L
U
L
L
A
I
L
L
A
C
O
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
1
-
-
-
1
-
-
-
-
1
occurs
x
2
=
2
=
2
-
3
3
3
3
3
-
-
3
3
-
3
--
-
-
3
occurs
x
8
=
24
2+4
6
-
-
-
-
-
-
-
-
-
-
-
-
6
-
-
6
occurs
x
1
=
6
=
6
-
-
-
-
-
-
-
9
-
-
-
-
-
-
-
9
occurs
x
1
=
9
=
9
12
L
L
U
L
L
A
I
L
L
A
C
O
-
-
19
-
-
12
-
41
-
23
1+2
3
3
3
3
3
-
-
3
3
-
3
--
-
-
1+9
-
-
1+2
-
4+1
-
2+3
3
L
L
U
L
L
A
I
L
L
A
C
O
-
-
10
-
-
3
-
5
-
5
-
3
3
3
3
3
1
9
3
3
1
3
6
-
-
1+0
-
-
-
-
-
-
-
3
L
L
U
L
L
A
I
L
L
A
C
O
-
-
1
-
-
3
-
5
-
5

 

 

The civilisation we discuss, which does not appear to have found a need to develop writing, is that of the Incas. The Inca empire which existed in 1532, ... www-history.mcs.st-andrews.ac.uk/ HistTopics/Inca_mathematics

History topic: Mathematics of the Incas

It is often thought that mathematics can only develop after a civilisation has developed some form of writing. Although not easy for us to understand today, many civilisations reached highly advanced states without ever developing written records. Now of course it is difficult for us to know much about such civilisations since there is no written record to be studied today. This article looks at the mathematical achievements of one such civilisation.

The civilisation we discuss, which does not appear to have found a need to develop writing, is that of the Incas. The Inca empire which existed in 1532, before the Spanish conquest, was vast. It spread over an area which stretched from what is now the northern border of Ecuador to Mendoza in west-central Argentina and to the Maule River in central Chile. The Inca people numbered around 12 million but they were from many different ethnic groups and spoke about 20 different languages. The civilisation had reached a high level of sophistication with a remarkable system of roads, agriculture, textile design, and administration. Of course even if writing is not required to achieve this level, counting and recording of numerical information is necessary. The Incas had developed a method of recording numerical information which did not require writing. It involved knots in strings called quipu.

The quipu was not a calculator, rather it was a storage device. Remember that the Incas had no written records and so the quipu played a major role in the administration of the Inca empire since it allowed numerical information to be kept. Let us first describe the basic quipu, with its positional number system, and then look at the ways that it was used in Inca society.

The quipu consists of strings which were knotted to represent numbers. A number was represented by knots in the string, using a positional base 10 representation. If the number 586 was to be recorded on the string then six touching knots were placed near the free end of the string, a space was left, then eight touching knots for the 10s, another space, and finally 5 touching knots for the 100s.

(Illustration omitted) 586 on a quipu.

For larger numbers more knot groups were used, one for each power of 10, in the same way as the digits of the number system we use here are occur in different positions to indicate the number of the corresponding power of 10 in that position.

Now it is not quite true that the same knots were used irrespective of the position as would be the case in a true positional system. There seems only one exception, namely the unit position, where different styles of knots were used from those in the other positions. In fact two different styles were used in the units position, one style if the unit were a 1 and a second style if the unit were greater than one. Both these styles differed from the standard knot used for all other positions. The system had a zero position, for this would be represented as no knots in that position. This meant that the spacing had to be highly regular so that zero positions would be clear.

There are many drawings and descriptions of quipus made by the Spanish invaders. Garcilaso de la Vega, whose mother was an Inca and whose father was Spanish, wrote (see for example [5]):-

According to their position, the knots signified units, tens, hundreds, thousands, ten thousands and, exceptionally, hundred thousands, and they are all well aligned on their different cords as the figures that an accountant sets down, column by column, in his ledger.

Now of course recording a number on a string would, in itself, not be that useful. A quipu had many strings and there had to be some way that the string carrying the record of a particular number could be identified. The primary way this was done was by the use of colour. Numbers were recorded on strings of a particular colour to identify what that number was recording. For example numbers of cattle might be recorded on green strings while numbers of sheep might be recorded on white strings. The colours each had several meanings, some of which were abstract ideas, some concrete as in the cattle and sheep example. White strings had the abstract meaning of "peace" while red strings had the abstract meaning of "war"

As well as the colour coding, another way of distinguishing the strings was to make some strings subsidiary ones, tied to the middle of a main string rather than being tied to the main horizontal cord.

(Illustration omitted) Quipu with subsidiary cords.

We quote Garcilaso de la Vega again [5]:-

The ordinary judges gave a monthly account of the sentences they imposed to their superiors, and they in turn reported to their immediate superiors, and so on finally to the Inca or those of his Supreme Council. The method of making these reports was by means of knots, made of various colours, where knots of such and such colours denote that such and such crimes had been punished. Smaller threads attached to thicker cords were of different colours to signify the precise nature of the punishment that had been inflicted.

It was not only judges who sent quipus to be kept in a central record. The Inca king appointed quipucamayocs, or keepers of the knots, to each town. Larger towns might have had up to thirty quipucamayocs who were essentially government statisticians, keeping official census records of the population, records of the produce of the town, its animals and weapons. This and other information was sent annually to the capital Cuzco. There was even an official delivery service to take to quipus to Cuzco which consisted of relay runners who passed the quipus on to the next runner at specially constructed staging posts. The terrain was extremely difficult yet the Incas had constructed roads to make the passing of information by quipus surprisingly rapid.

Much information on the quipus comes from a letter of the Peruvian Felipe Guaman Poma de Ayala to the King of Spain, written about eighty years after the Spanish conquest of the Incas. This remarkable letter contains 1179 pages and there are several drawings which show quipus. A fascinating aspect of one of these drawing is a picture of a counting board in the bottom left hand corner of one of them. This is called the yupana and is presumed to be the counting board of the Incas.

(Illustration omitted) This is what the yupana looked like.

Interpretations of how this counting board, or Peruvian abacus, might have been used have been given by several authors, see for example [9] and [11]. However some historians are less certain that this really is a Peruvian abacus. For example [2] in which the authors write:-

It is unclear from Poma's commentary whether it is his version of a device associated with Spanish activities analogous to those of the person depicted or whether he is implying its association with the Incas. In either case, his commentary makes interpretation of the configuration and the meaning of the unfilled and filled holes highly speculative.

It is a difficult task to gain further insights into the mathematical understanding of the Incas. The book [6] by Urton is interesting for it examines the concept of number as understood by the Inca people. As one might expect, their concept of number was a very concrete one, unlike our concept of number which is a highly abstract one (although this is not really understood by many people). The concrete way of conceiving numbers is illustrated by different words used when describing properties of numbers. One example given in [6] is that of even and odd numbers. Now the ideas of an even number, say, relies on having an abstract concept of number which is independent of the objects being counted. However, the Peruvian languages had different words which applied to different types of objects. For example separate words occur for the idea of [6]:-

... the two together that make a pair ...

... the one together with its mate ...

... two - in reference to one thing that is divided into two parts ...

... a pair of two separate things bound intimately together, such as two bulls yoked together for ploughing ... etc.

This is a fascinating topic and one which deserves much further research. One wonders whether the Incas applied their number system to solve mathematical problems. Was it merely for recording? If the yupana really was an abacus then it must have been used to solve problems and this prompts the intriguing question of what these problems were. A tantalising glimpse may be contained in the writings of the Spanish priest José de Acosta who lived among the Incas from 1571 to 1586. He writes in his book Historia Natural Moral de las Indias which was published in Madrid in 1596:-

To see them use another kind of calculator, with maize kernels, is a perfect joy. In order to carry out a very difficult computation for which an able computer would require pen and paper, these Indians make use of their kernels. They place one here, three somewhere else and eight, I know not where. They move one kernel here and there and the fact is that they are able to complete their computation without making the smallest mistake. as a matter of fact, they are better at practical arithmetic than we are with pen and ink. Whether this is not ingenious and whether these people are wild animals let those judge who will! What I consider ascertain is that in what they undertake to do they are superior to us.

What a pity that de Acosta did not have the mathematical skills to give a precise description which would have allowed us to understand this method of calculation by the Incas.

Article by: J J O'Connor and E F Robertson January 2001 MacTutor History of Mathematics

[http://www-history.mcs.st-andrews.ac.uk/HistTopics/Inca_mathematics.html]

 

 

I

INCA

THE

SON OF THE SUN

 

 

ENUMA ELISH - Babylonian Creation Myth - The continued story www.stenudd.com/myth/enumaelish/enumaelish-


The word used for man is lullu, meaning a first, primitive man. The same word is used about the savage Enkidu in the Gilgamesh epic. Since Qingu is found ...

I hereby name it Babylon, home of the great gods.

The word used in the text is written phonetically, ba-ab-i-li, contrary to tradition, maybe to allow for the etymological explanation of the name as the ‘gate of the gods’.
Then he decides to create man, to serve the gods with offerings, so that they can be at leisure. The word used for man is lullu, meaning a first, primitive man. The same word is used about the savage Enkidu in the Gilgamesh epic. Since Qingu is found guilty of the war between the gods, his blood is used to create mankind. Here, it is unclear if Marduk or Ea creates mankind. Later in the text, Ea is specified as the creator of man. Finally, the gods praise Marduk, and give him fifty names that represent different aspects of his powers and sovereignty.
The text ends with instructions on how it should be passed on from generation to generation, and the command to worship Marduk, king of the gods.

ENUMA ELISH
The Babylonian Creation Myth

 

"The word used for man is lullu"

LULLU 33333 LULLU

"The word used for man is lullu"

 

 

10
CURRICULUM
--
-
-
-
C
3
3
3
-
U
21
3
3
-
R
18
9
9
-
R
18
9
9
-
I
9
9
9
-
C
3
3
3
-
U
21
3
3
-
L
12
3
3
-
U
21
3
3
-
M
13
4
4
10
CURRICULUM
139
49
49
1+0
-
1+3+9
4+9
4+9
1
CURRICULUM
13
13
13
-
-
1+3
1+3
1+3
1
CURRICULUM
4
4
4

 

 

10
CURRICULUM
--
-
-
-
C
3
3
3
-
U
21
3
3
-
R
18
9
9
-
R
18
9
9
-
I
9
9
9
-
C
3
3
3
-
U
21
3
3
-
L
12
3
3
-
U
21
3
3
-
M
13
4
4
10
CURRICULUM
139
49
49
1+0
-
1+3+9
4+9
4+9
1
CURRICULUM
13
13
13
-
-
1+3
1+3
1+3
1
CURRICULUM
4
4
4

 

 

10
CURRICULUM
--
-
-
-
-
-
-
-
C
3
3
3
-
3
-
-
-
U
21
3
3
-
3
-
-
-
R
18
9
9
-
-
-
9
-
R
18
9
9
-
-
-
9
-
I
9
9
9
-
-
-
9
-
C
3
3
3
-
3
-
-
-
U
21
3
3
-
3
-
-
-
L
12
3
3
-
3
-
-
-
U
21
3
3
-
3
-
-
-
M
13
4
4
-
-
4
-
10
CURRICULUM
139
49
49
-
18
4
27
1+0
-
1+3+9
4+9
4+9
-
1+8
-
2+7
1
CURRICULUM
13
13
13
-
9
4
9
-
-
1+3
1+3
1+3
-
-
-
-
1
CURRICULUM
4
4
4
-
9
4
9

 

 

-
10
C
U
R
R
I
C
U
L
U
M
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
9
-
-
-
-
-
+
=
9
-
=
9
-
9
=
9
-
-
-
-
-
-
9
-
-
-
-
-
+
=
9
-
=
9
-
9
=
9
-
10
C
U
R
R
I
C
U
L
U
M
-
-
-
-
-
-
-
-
-
-
-
-
3
3
9
9
-
3
3
3
3
4
+
=
40
4+0
=
4
=
4
=
4
-
-
3
21
18
18
-
3
21
12
21
13
+
=
130
1+3+0
=
4
=
4
=
4
-
10
C
U
R
R
I
C
U
L
U
M
-
-
-
-
-
-
-
-
-
-
-
-
3
21
18
18
9
3
21
12
21
13
+
=
139
1+3+9
=
13
1+3
4
=
4
-
-
3
3
9
9
9
3
3
3
3
4
+
=
49
4+9
=
13
1+3
4
=
4
-
10
C
U
R
R
I
C
U
L
U
M
-
-
-
-
-
-
-
-
-
-
1
-
-
-
-
-
-
-
-
-
-
-
-
-
1
ONE
1
-
-
-
-
-
2
-
-
-
-
-
-
-
-
-
-
-
-
-
2
TWO
2
-
-
-
-
-
-
-
3
3
-
-
-
3
3
3
3
-
-
-
3
occurs
x
6
=
18
1+8
9
-
-
-
-
-
-
-
-
-
-
-
4
-
-
4
occurs
x
1
=
4
=
4
5
-
-
-
-
-
-
-
-
-
-
-
-
-
5
FIVE
5
-
-
-
-
-
6
-
-
-
-
-
-
-
-
-
-
-
-
-
6
SIX
6
-
-
-
-
-
7
-
-
-
-
-
-
-
-
-
-
-
-
-
7
SEVEN
7
-
-
-
-
-
8
-
-
-
-
-
-
-
-
-
-
-
-
-
8
EIGHT
8
-
-
-
-
-
-
-
-
-
9
9
9
-
-
-
-
-
-
-
9
occurs
x
3
=
27
2+7
9
29
10
C
U
R
R
I
C
U
L
U
M
-
-
16
-
-
10
-
49
-
22
2+9
1+0
3
3
-
-
-
3
3
3
3
-
-
-
1+6
-
-
1+0
-
4+9
-
2+2
11
1
C
U
R
R
I
C
U
L
U
M
-
-
7
-
-
1
-
13
-
4
1+1
-
3
3
9
9
9
3
3
3
3
4
-
-
-
-
-
-
-
1+3
-
-
2
1
C
U
R
R
I
C
U
L
U
M
-
-
7
-
-
1
-
4
-
4

 

 

10
C
U
R
R
I
C
U
L
U
M
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
9
-
-
-
-
-
+
=
9
-
=
9
-
9
=
9
-
-
-
-
-
9
-
-
-
-
-
+
=
9
-
=
9
-
9
=
9
10
C
U
R
R
I
C
U
L
U
M
-
-
-
-
-
-
-
-
-
-
-
3
3
9
9
-
3
3
3
3
4
+
=
40
4+0
=
4
=
4
=
4
-
3
21
18
18
-
3
21
12
21
13
+
=
130
1+3+0
=
4
=
4
=
4
10
C
U
R
R
I
C
U
L
U
M
-
-
-
-
-
-
-
-
-
-
-
3
21
18
18
9
3
21
12
21
13
+
=
139
1+3+9
=
13
1+3
4
=
4
-
3
3
9
9
9
3
3
3
3
4
+
=
49
4+9
=
13
1+3
4
=
4
10
C
U
R
R
I
C
U
L
U
M
-
-
-
-
-
-
-
-
-
-
-
3
3
-
-
-
3
3
3
3
-
-
-
3
occurs
x
6
=
18
1+8
9
-
-
-
-
-
-
-
-
-
-
4
-
-
4
occurs
x
1
=
4
=
4
-
-
-
9
9
9
-
-
-
-
-
-
-
9
occurs
x
3
=
27
2+7
9
10
C
U
R
R
I
C
U
L
U
M
-
-
16
-
-
10
-
49
-
22
1+0
3
3
-
-
-
3
3
3
3
-
-
-
1+6
-
-
1+0
-
4+9
-
2+2
1
C
U
R
R
I
C
U
L
U
M
-
-
7
-
-
1
-
13
-
4
-
3
3
9
9
9
3
3
3
3
4
-
-
-
-
-
-
-
1+3
-
-
1
C
U
R
R
I
C
U
L
U
M
-
-
7
-
-
1
-
4
-
4

 

 

11
7
C
A
L
C
U
L
I
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
9
+
=
9
-
=
9
-
9
=
9
-
-
-
-
-
-
-
-
9
+
=
9
-
=
9
-
9
=
9
-
7
C
A
L
C
U
L
I
-
-
-
-
-
-
-
-
-
-
-
-
3
1
3
3
3
3
-
+
=
16
1+6
=
7
-
7
=
7
-
-
3
1
12
3
21
12
-
+
=
52
5+2
=
7
-
7
=
7
11
7
C
A
L
C
U
L
I
-
-
-
-
-
-
-
-
-
-
-
-
3
1
12
3
21
12
9
+
=
61
6+1
=
7
-
7
=
7
-
-
3
1
3
3
3
3
9
+
=
25
2+5
=
7
-
7
=
7
11
7
C
A
L
C
U
L
I
-
-
-
-
-
-
-
-
-
-
-
-
-
1
-
-
-
-
-
-
-
1
occurs
x
1
=
1
=
1
2
-
-
-
-
-
-
-
-
-
-
2
TWO
-
-
-
-
-
-
-
-
3
-
3
3
3
3
-
-
-
3
occurs
x
5
=
15
1+5
6
4
-
-
-
-
-
-
-
-
-
-
4
FOUR
4
-
-
-
-
-
5
-
-
-
-
-
-
-
-
-
-
5
FIVE
5
-
-
-
-
-
6
-
-
-
-
-
-
-
-
-
-
6
SIX
6
-
-
-
-
-
7
-
-
-
-
-
-
-
-
-
-
7
SEVEN
7
-
-
-
-
-
8
-
-
-
-
-
-
-
-
-
-
8
EIGHT
8
-
-
-
-
-
-
-
-
-
-
-
-
-
9
-
-
9
occurs
x
1
=
9
=
9
32
7
C
A
L
C
U
L
I
-
-
13
-
-
7
-
25
-
16
3+2
-
3
-
3
3
3
3
-
-
-
1+3
-
-
-
-
2+5
-
1+6
5
7
C
A
L
C
U
L
I
-
-
4
-
-
7
-
7
-
7
-
-
3
1
3
3
3
3
9
-
-
-
-
-
-
-
-
-
-
5
7
C
A
L
C
U
L
I
-
-
4
-
-
7
-
7
-
7

 

 

7
C
A
L
C
U
L
I
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
9
+
=
9
-
=
9
-
9
=
9
-
-
-
-
-
-
-
9
+
=
9
-
=
9
-
9
=
9
7
C
A
L
C
U
L
I
-
-
-
-
-
-
-
-
-
-
-
3
1
3
3
3
3
-
+
=
16
1+6
=
7
-
7
=
7
-
3
1
12
3
21
12
-
+
=
52
5+2
=
7
-
7
=
7
7
C
A
L
C
U
L
I
-
-
-
-
-
-
-
-
-
-
-
3
1
12
3
21
12
9
+
=
61
6+1
=
7
-
7
=
7
-
3
1
3
3
3
3
9
+
=
25
2+5
=
7
-
7
=
7
7
C
A
L
C
U
L
I
-
-
-
-
-
-
-
-
-
-
-
-
1
-
-
-
-
-
-
-
1
occurs
x
1
=
1
=
1
-
3
-
3
3
3
3
-
-
-
3
occurs
x
5
=
15
1+5
6
-
-
-
-
-
-
-
9
-
-
9
occurs
x
1
=
9
=
9
7
C
A
L
C
U
L
I
-
-
13
-
-
7
-
25
-
16
-
3
-
3
3
3
3
-
-
-
1+3
-
-
-
-
2+5
-
1+6
7
C
A
L
C
U
L
I
-
-
4
-
-
7
-
7
-
7
-
3
1
3
3
3
3
9
-
-
-
-
-
-
-
-
-
-
7
C
A
L
C
U
L
I
-
-
4
-
-
7
-
7
-
7

 

 

5
COMET
56
20
2
5
LULIN
56
20
2
1
L
12
3
3
1
U
21
3
3
1
L
12
3
3
1
I
9
9
9
1
N
14
5
5
5
LULIN
68
23
23
-
-
6+8
2+3
2+3
5
LULIN
14
5
5
-
-
1+4
-
-
5
LULIN
5
5
5

 

 

5
LULIN
56
20
2
3
L+U+L
45
9
9
1
I
9
9
9
1
N
14
5
5
5
LULIN
-
-
-

 

 

C RE ATUM ATUM RE C

SEE ATUM RE ATUM RE SEE

C RE ATUM ATUM RE C

CREATION SEE REACTION

CREATORS SEE CREATORS

CREATIVE SEE REACTIVE SEE CREATIVE

CREATUM 3951234 CREATUM

CREATUM

3951234

CREATUM

 

 

OSIRIS ISISIS OSIRIS

ISISIS

OSIRIS ISISISISISIS SOIRIS

IRIS OSIRIS ISIS

SO IRIS ISIS SO

OSIRIS ISISIS SO IR IS OSIRIS ISISIS

SO IR IS 99 IS IR SO

?

 

 

From album "Strip" (2000)

I Must have died a thousand times
I Must have died a thousand times
I Must have died a thousand times
Feeling less than human
I surmise
I'm less than human in God's eyes

Time is on my side she said
Time is on my side she said
He may be on your side I said,
But it makes no difference in the end
He's coming after you my friend

I Must have cried a thousand times
I Must have cried a thousand times
I Must have cried a thousand times
Feeling less than human
I surmise
I'm less than human in God's eyes

How time flies
But time is on my side, She said

Oh no no no no, oh no no no no

 

 

JUST SIX NUMBERS

Martin Rees

1
999

OUR COSMIC HABITAT

PLANETS STARS AND LIFE

Page 24

A

proton

is

1,836 times heavier than an electron, and the number 1,836

would have the same connotations to any 'intelligence'

 

 

-
-
-
-
-
LULLABY
-
-
-
-
-
-
-
1
L
12
3
3
-
-
-
-
1
U
21
3
3
-
-
-
-
1
L
12
3
3
-
-
-
-
1
L
12
3
3
-
-
-
-
2
A+B
3
3
3
-
-
-
-
1
Y
25
7
7
L
=
3
-
7
LULLABY
85
22
22
-
-
-
-
-
-
8+5
2+2
2+2
L
=
3
-
7
LULLABY
13
4
4
-
-
-
-
-
-
1+3
-
-
L
=
3
-
7
LULLABY
4
4
4

 

 

EVOLVE LOVE LOVE EVOLVE

 

-
EVOLVE LOVE LOVE EVOLVE
-
-
-
-
EVOLVE
-
-
-
2
E+V
27
9
9
2
O+L
27
9
9
2
V+E
27
9
9
-
EVOLVE
-
-
-
-
LOVE
-
-
-
2
L+O
27
9
9
2
V+E
27
9
9
-
LOVE
-
-
-
-
LOVE
-
-
-
2
L+O
27
9
9
2
V+E
27
9
9
-
LOVE
-
-
-
-
EVOLVE
-
-
-
2
E+V
27
9
9
2
O+L
27
9
9
2
V+E
27
9
9
-
EVOLVE
-
-
-
20
EVOLVE LOVE LOVE EVOLVE
270
90
45
2+0
-
2+7+0
9+0
4+5
2
EVOLVE LOVE LOVE EVOLVE
9
9
9
-
-
-
-
-
2
EVOLVE LOVE LOVE EVOLVE
9
9
9

 

REVOLVER EVOLVER LOVER

 

-
-
-
-
-
REVOLVER
-
-
-
-
-
-
-
1
R
18
9
9
-
-
-
-
2
E+V
27
9
9
-
-
-
-
2
O+L
27
9
9
-
-
-
-
2
V+E
27
9
9
-
-
-
-
1
R
18
9
9
R
=
9
-
8
REVOLVER
117
45
45
-
-
-
-
-
EVOLVER
-
-
-
-
-
-
-
2
E+V
27
9
9
-
-
-
-
2
O+L
27
9
9
-
-
-
-
2
V+E
27
9
9
-
-
-
-
1
R
18
9
9
E
=
5
-
7
EVOLVER
99
36
36
-
-
-
-
-
LOVER
-
-
-
-
5
-
-
2
L+O
27
9
9
-
1
-
-
2
V+E
27
9
9
-
1
-
-
1
R
18
9
9
L
=
3
-
5
LOVER
72
27
27
-
-
-
-
-
-
-
-
-
-
-
17
-
20
288
108
108
-
-
1+7
-
2+0
-
2+7+0
1+0+8
1+0+8
-
-
8
-
2
9
9
9

 

REVOLVER EVOLVER LOVER

 

 

T
=
1
-
2
TO
35
8
8
B
=
2
-
2
BE
7
7
7
O
=
6
-
2
OR
33
15
6
N
=
1
-
3
NOT
49
13
4
T
=
1
-
2
TO
35
8
8
B
=
2
-
2
BE
7
7
7
T
=
2
-
4
THAT
49
13
4
I
=
9
-
2
IS
28
10
1
T
=
3
-
3
THE
33
15
6
Q
=
8
-
8
QUESTION
120
39
3
-
-
36
-
30
-
396
135
54
-
-
3+6
-
3+0
-
3+9+6
1+3+5
5+4
Q
-
9
Q
3
Q
18
9
9
-
-
-
-
-
-
1+8
-
-
-
-
9
-
3
-
9
9
9

 

 

 
 
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